VSET presentation slides presented on June 13 online

VSET presentation slides presented on June 13 online

1. A mechanism-design approach to property rights
Piotr Dworczak
?
Ellen Muir
(Northwestern; GRAPE) (Harvard)
June 13, 2024
Virtual Seminar in Economic Theory (VSET)
?
Co-funded by the European Union (ERC, IMD-101040122). Views and opinions expressed are those of the authors
only and do not necessarily reect those of the European Union or the European Research Council.

2. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretrade-offsin the design of property rights:
Investment incentives;
Efciency of reallocation;
Market power and distribution of surplus.
How to optimally design property rights?

3. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretrade-offsin the design of property rights:
Investment incentives;
Efciency of reallocation;
Market power and distribution of surplus.
How to optimally design property rights?

4. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretrade-offsin the design of property rights:
Investment incentives;
Efciency of reallocation;
Market power and distribution of surplus.
How to optimally design property rights?

5. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretrade-offsin the design of property rights:
Investment incentives;
Efciency of reallocation;
Market power and distribution of surplus.
Example: How to optimally design spectrum licenses?

6. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

7. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

8. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

9. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

10. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

11. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

12. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

13. Our framework
Property rightsdetermine the holder's outside options(with
regard to the underlying good) in economic interactions:
Full property right: Holder can always keep the good;
No right: Outside option is (normalized to) zero;Option to own: Holder can keep the good at a price....
Key modeling idea: Designerdoesn't control the allocation
directly but shecan design an agent's outside option.
This perspective allows us to characterize theoptimal property
rightusing a mechanism-design approach.
Result: The optimal property right is simple but more exible than
a conventional property right, often featuring anoption to own.

14. Marginal contribution
Two large literatures studying property rights in the presence of
transaction costs:
Property rights affectefciencyunder private information:
Myerson and Satterthwaite (1983), Cramton, Gibbons and
Klemperer (1987),Che (2006), Segal and Whinston (2016), ...
Property rights affectinvestment incentives:
Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994),Hart (1995), N¨oldeke and Schmidt
(1995, 1998), Maskin and Tirole (1999b), ...
Attractive properties of options to own were recognized.
We provide anoptimality foundation(in a “refreshed”
framework, allowing for market-design applications).

15. Marginal contribution
Two large literatures studying property rights in the presence of
transaction costs:
Property rights affectefciencyunder private information:
Myerson and Satterthwaite (1983), Cramton, Gibbons and
Klemperer (1987),Che (2006), Segal and Whinston (2016), ...
Property rights affectinvestment incentives:
Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994),Hart (1995), N¨oldeke and Schmidt
(1995, 1998), Maskin and Tirole (1999b), ...
Attractive properties of options to own were recognized.
We provide anoptimality foundation(in a “refreshed”
framework, allowing for market-design applications).

16. Marginal contribution
Two large literatures studying property rights in the presence of
transaction costs:
Property rights affectefciencyunder private information:
Myerson and Satterthwaite (1983), Cramton, Gibbons and
Klemperer (1987),Che (2006), Segal and Whinston (2016), ...
Property rights affectinvestment incentives:
Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994),Hart (1995), N¨oldeke and Schmidt
(1995, 1998), Maskin and Tirole (1999b), ...
Attractive properties of options to own were recognized.
We provide anoptimality foundation(in a “refreshed”
framework, allowing for market-design applications).

17. Marginal contribution
Two large literatures studying property rights in the presence of
transaction costs:
Property rights affectefciencyunder private information:
Myerson and Satterthwaite (1983), Cramton, Gibbons and
Klemperer (1987),Che (2006), Segal and Whinston (2016), ...
Property rights affectinvestment incentives:
Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994),Hart (1995), N¨oldeke and Schmidt
(1995, 1998), Maskin and Tirole (1999b), ...
Attractive properties of options to own were recognized.
We provide anoptimality foundation(in a “refreshed”
framework, allowing for market-design applications).

18. Other related papers
Allocation mechanisms versus efcient investment:
Rogerson (1992), Bergemann and V¨alim¨aki (2002), Milgrom
(2017), Hateld, Kojima, and Kominers (2019), Gershkov,
Moldovanu, Strack, and Zhang (2021), Akbarpour, Kominers, Li,
Li, and Milgrom (2023), ...
Related techniques:Jullien (2000), Kleiner, Moldovanu, and
Strack (2021), Loertscher and Muir (2022), Kang (2023),
Akbarpour
R
Dworczak
R
Kominers (2023),...
Spectrum license design:Posner and Weyl (2017), Milgrom,
Weyl and Zhang (2017), Weyl and Zhang (2017), ...
Optimal patent design:Wright (1983), Klemperer (1990),
Gilbert and Shapiro (1990), Gallini (1992), Kremer (1998),
Hopenhayn, Llobet and Mitchell (2006), Weyl and Tirole (2012)...

19. Outline
1.
2.
3.
4.
5.
6.

20. Model
Model
[Simplied version without investment]

21. Model
Standard contracting problem between principal and agent:
Public state!is drawn from distributionG.Agent's private typeis drawn from distributionF!.
Principalchooses a trading mechanism(x!();t!())subject to
IC and IR constraints, wherex2[0;1]denotes an allocation,
andt2Rdenotes a transfer.
Agent's utility isxt.
Principal maximizesV(; !)x+t, where >0.

22. Model
Standard contracting problem between principal and agent:
Public state!is drawn from distributionG.Agent's private typeis drawn from distributionF!.
Principalchooses a trading mechanism(x!();t!())subject to
IC and IR constraints, wherex2[0;1]denotes an allocation,
andt2Rdenotes a transfer.
Agent's utility isxt.
Principal maximizesV(; !)x+t, where >0.

23. Model
Standard contracting problem between principal and agent:
Public state!is drawn from distributionG.Agent's private typeis drawn from distributionF!.
Principalchooses a trading mechanism(x!();t!())subject to
IC and IR constraints, wherex2[0;1]denotes an allocation,
andt2Rdenotes a transfer.
Agent's utility isxt.
Principal maximizesV(; !)x+t, where >0.

24. Model
Standard contracting problem between principal and agent:
Public state!is drawn from distributionG.Agent's private typeis drawn from distributionF!.
Principalchooses a trading mechanism(x!();t!())subject to
IC and IR constraints, wherex2[0;1]denotes an allocation,
andt2Rdenotes a transfer.
Agent's utility isxt.
Principal maximizesV(; !)x+t, where >0.

25. Model
Standard contracting problem between principal and agent:
Public state!is drawn from distributionG.Agent's private typeis drawn from distributionF!.
Principalchooses a trading mechanism(x!();t!())subject to
IC and IR constraints, wherex2[0;1]denotes an allocation,
andt2Rdenotes a transfer.
Agent's utility isxt.
Principal maximizesV(; !)x+t, where >0.

26. Model
Before the interaction of the agent with the principal,
a designer (6=principal) species the agent's property rights.
Designerchooses a menu of outside options
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary
(assumeMis compact).
The agent can “execute” any one of these rights.Designer's payoff is given byV
?
(; !)x+
?
t, where
?
0.Assume:[;]is compact;VandV
?
are continuous inand
measurable in!,F!has a continuous positive density on.

27. Model
Before the interaction of the agent with the principal,
a designer (6=principal) species the agent's property rights.
Designerchooses a menu of outside options
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary
(assumeMis compact).
The agent can “execute” any one of these rights.Designer's payoff is given byV
?
(; !)x+
?
t, where
?
0.Assume:[;]is compact;VandV
?
are continuous inand
measurable in!,F!has a continuous positive density on.

28. Model
Before the interaction of the agent with the principal,
a designer (6=principal) species the agent's property rights.
Designerchooses a menu of outside options
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary
(assumeMis compact).
The agent can “execute” any one of these rights.Designer's payoff is given byV
?
(; !)x+
?
t, where
?
0.Assume:[;]is compact;VandV
?
are continuous inand
measurable in!,F!has a continuous positive density on.

29. Model
Before the interaction of the agent with the principal,
a designer (6=principal) species the agent's property rights.
Designerchooses a menu of outside options
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary
(assumeMis compact).
The agent can “execute” any one of these rights.Designer's payoff is given byV
?
(; !)x+
?
t, where
?
0.Assume:[;]is compact;VandV
?
are continuous inand
measurable in!,F!has a continuous positive density on.

30. Model
Before the interaction of the agent with the principal,
a designer (6=principal) species the agent's property rights.
Designerchooses a menu of outside options
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary
(assumeMis compact).
The agent can “execute” any one of these rights.Designer's payoff is given byV
?
(; !)x+
?
t, where
?
0.Assume:[;]is compact;VandV
?
are continuous inand
measurable in!,F!has a continuous positive density on.

31. Model
Before the interaction of the agent with the principal,
a designer (6=principal) species the agent's property rights.
Designerchooses a menu of outside options
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary
(assumeMis compact).
The agent can “execute” any one of these rights.Designer's payoff is given byV
?
(; !)x+
?
t, where
?
0.Assume:[;]is compact;VandV
?
are continuous inand
measurable in!,F!has a continuous positive density on.

32. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

33. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

34. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

35. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

36. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

37. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

38. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

39. Comments about the model
Rights create anoutside optionfor the agent: principal must
guarantee that the agent's type-utility from participating in the
mechanism is not lower thanmax
i2Ifx
it
ig.
The framework captures many conventional rights:Property right:M=f(x=1;t=0)g;
Cash payment:M=f(0;p)g;Property right with a resale option:M=f(1;0);(0;p)g;Renewable lease/ option to own:M=f(1;p)g;Partial property right:M=f(y;0)g, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

40. Comments about the model
We assumed that property rights cannot be made contingent on
the state!(“incomplete contracts”);Straightforward extension to the case when property rights can
depend on some “contractible state”s, jointly distributed with!.
Uninteresting cases:
If the designer and principal have aligned preferences, it is optimal
to allocate no rights to the agent.If the principal did not care about revenue (=0), she would “buy
out” the agent's rights.

41. Comments about the model
We assumed that property rights cannot be made contingent on
the state!(“incomplete contracts”);Straightforward extension to the case when property rights can
depend on some “contractible state”s, jointly distributed with!.
Uninteresting cases:
If the designer and principal have aligned preferences, it is optimal
to allocate no rights to the agent.If the principal did not care about revenue (=0), she would “buy
out” the agent's rights.

42. Comments about the model
We assumed that property rights cannot be made contingent on
the state!(“incomplete contracts”);Straightforward extension to the case when property rights can
depend on some “contractible state”s, jointly distributed with!.
Uninteresting cases:
If the designer and principal have aligned preferences, it is optimal
to allocate no rights to the agent.If the principal did not care about revenue (=0), she would “buy
out” the agent's rights.

43. Comments about the model
We assumed that property rights cannot be made contingent on
the state!(“incomplete contracts”);Straightforward extension to the case when property rights can
depend on some “contractible state”s, jointly distributed with!.
Uninteresting cases:
If the designer and principal have aligned preferences, it is optimal
to allocate no rights to the agent.If the principal did not care about revenue (=0), she would “buy
out” the agent's rights.

44. Comments about the model
We assumed that property rights cannot be made contingent on
the state!(“incomplete contracts”);Straightforward extension to the case when property rights can
depend on some “contractible state”s, jointly distributed with!.
Uninteresting cases:
If the designer and principal have aligned preferences, it is optimal
to allocate no rights to the agent.If the principal did not care about revenue (=0), she would “buy
out” the agent's rights.

45. Analysis and results
Analysis and results

46. Main result
Theorem
There exists an optimal menu that takes the form M
?
=f(1;p)gfor
some p2R.
Comments:
The optimal menu is asingleton.
The unique outside option offered is anoption to own.Disclaimer: The menu will get more complicated once we add
the hold-up problem.

47. Main result
Theorem
There exists an optimal menu that takes the form M
?
=f(1;p)gfor
some p2R.
Comments:
The optimal menu is asingleton.
The unique outside option offered is anoption to own.Disclaimer: The menu will get more complicated once we add
the hold-up problem.

48. Main result
Theorem
There exists an optimal menu that takes the form M
?
=f(1;p)gfor
some p2R.
Comments:
The optimal menu is asingleton.
The unique outside option offered is anoption to own.Disclaimer: The menu will get more complicated once we add
the hold-up problem.

49. Main result
Theorem
There exists an optimal menu that takes the form M
?
=f(1;p)gfor
some p2R.
Comments:
The optimal menu is asingleton.
The unique outside option offered is anoption to own.Disclaimer: The menu will get more complicated once we add
the hold-up problem.

50. Step 1: Outside option constraint
Lemma
A choice of menu M by the designer is equivalent to choosing an
outside option function R()for the agent in the mechanism, where
R()is non-negative, non-decreasing, convex, and has slope
bounded above by1.
The lemma follows immediately from the observation that, for anyM,
we can set
R() = maxf0;max
i2I
fx
it
igg:
The designermaximizes over type-dependent outside optionsfor
the agent.

51. Step 1: Outside option constraint
Lemma
A choice of menu M by the designer is equivalent to choosing an
outside option function R()for the agent in the mechanism, where
R()is non-negative, non-decreasing, convex, and has slope
bounded above by1.
The lemma follows immediately from the observation that, for anyM,
we can set
R() = maxf0;max
i2I
fx
it
igg:
The designermaximizes over type-dependent outside optionsfor
the agent.

52. Step 1: Outside option constraint
Lemma
A choice of menu M by the designer is equivalent to choosing an
outside option function R()for the agent in the mechanism, where
R()is non-negative, non-decreasing, convex, and has slope
bounded above by1.
The lemma follows immediately from the observation that, for anyM,
we can set
R() = maxf0;max
i2I
fx
it
igg:
The designermaximizes over type-dependent outside optionsfor
the agent.

53. Step 1: Outside option constraint
Fixing!(and dropping it from the notation), the principal solves:
max
x:![0;1];u0
Z


W()x()du
s.t.xis non-decreasing;
U()u+
Z


x()dR();82;
where
W()(V() +B())f();
and
B() =
1F()
f()
:
This problem was considered by Jullien (2000).

54. Step 1: Outside option constraint
Fixing!(and dropping it from the notation), the principal solves:
max
x:![0;1];u0
Z


W()x()du
s.t.xis non-decreasing;
U()u+
Z


x()dR();82;
where
W()(V() +B())f();
and
B() =
1F()
f()
:
This problem was considered by Jullien (2000).

55. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

56. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

57. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

58. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

59. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

60. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

61. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

62. Step 2: Ironing Jullien (2000)

63. Step 2: Ironing Jullien (2000)

64. Step 2: Ironing Jullien (2000)

65. Step 2: Ironing Jullien (2000)

66. Step 2: Ironing Jullien (2000)

67. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

68. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

69. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

70. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

71. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

72. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

73. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

74. Step 3: Linearity of the designer's problem
We now go back to the designer's problem.
Lemma (Linearity)
The designer's problem of choosing the optimal menu M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
The designer's objective is linear inx
?
.

75. Step 3: Linearity of the designer's problem
We now go back to the designer's problem.
Lemma (Linearity)
The designer's problem of choosing the optimal menu M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
The designer's objective is linear inx
?
.

76. Step 3: Linearity of the designer's problem
We now go back to the designer's problem.
Lemma (Linearity)
The designer's problem of choosing the optimal menu M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
The designer's objective is linear inx
?
.

77. Step 3: Linearity of the designer's problem
We now go back to the designer's problem.
Lemma (Linearity)
The designer's problem of choosing the optimal menu M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
The designer's objective is linear inx
?
.

78. Step 3: Linearity of the designer's problem
We now go back to the designer's problem.
Lemma (Linearity)
The designer's problem of choosing the optimal menu M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
The designer's objective is linear inx
?
.

79. Step 4: Wrapping up
Lemma
The designer's problem of choosing the optimal menu M is equivalent
to choosing a probability distribution and a scalar to maximize a linear
objective over a convex set.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution that is an extreme point:
x0corresponds to a degenerate distribution.
This gives us the form of the optimal menu.

80. Step 4: Wrapping up
Lemma
The designer's problem of choosing the optimal menu M is equivalent
to choosing a probability distribution and a scalar to maximize a linear
objective over a convex set.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution that is an extreme point:
x0corresponds to a degenerate distribution.
This gives us the form of the optimal menu.

81. Step 4: Wrapping up
Lemma
The designer's problem of choosing the optimal menu M is equivalent
to choosing a probability distribution and a scalar to maximize a linear
objective over a convex set.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution that is an extreme point:
x0corresponds to a degenerate distribution.
This gives us the form of the optimal menu.

82. Step 4: Wrapping up
Lemma
The designer's problem of choosing the optimal menu M is equivalent
to choosing a probability distribution and a scalar to maximize a linear
objective over a convex set.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution that is an extreme point:
x0corresponds to a degenerate distribution.
This gives us the form of the optimal menu.

83. Illustration
Illustration

84. Illustration
Principal is a seller of an indivisible good, maximizingrevenue.
State!represents seller's opportunitycostof allocating the
good to the agent, andv(!)represents the (expected) social
opportunity cost.
Typeis the buyer'svaluefor the good.Designer cares about allocativeefciency
(First best: VCG mechanism charging a pricev(!)to the buyer).

85. Illustration
Principal is a seller of an indivisible good, maximizingrevenue.
State!represents seller's opportunitycostof allocating the
good to the agent, andv(!)represents the (expected) social
opportunity cost.
Typeis the buyer'svaluefor the good.Designer cares about allocativeefciency
(First best: VCG mechanism charging a pricev(!)to the buyer).

86. Illustration
Principal is a seller of an indivisible good, maximizingrevenue.
State!represents seller's opportunitycostof allocating the
good to the agent, andv(!)represents the (expected) social
opportunity cost.
Typeis the buyer'svaluefor the good.Designer cares about allocativeefciency
(First best: VCG mechanism charging a pricev(!)to the buyer).

87. Illustration
Principal is a seller of an indivisible good, maximizingrevenue.
State!represents seller's opportunitycostof allocating the
good to the agent, andv(!)represents the (expected) social
opportunity cost.
Typeis the buyer'svaluefor the good.Designer cares about allocativeefciency
(First best: VCG mechanism charging a pricev(!)to the buyer).

88. Illustration
By the Theorem, the best the designer can do is endow the
agent with anoption to ownwith some pricep.
That is, the agent has the option to pay the pricepand obtain
the good.
Whenpis low, this approaches a full property right; whenpis
high, this approaches no right.
We can say more about the optimalpunder further regularity
conditions.

89. Illustration
By the Theorem, the best the designer can do is endow the
agent with anoption to ownwith some pricep.
That is, the agent has the option to pay the pricepand obtain
the good.
Whenpis low, this approaches a full property right; whenpis
high, this approaches no right.
We can say more about the optimalpunder further regularity
conditions.

90. Illustration
By the Theorem, the best the designer can do is endow the
agent with anoption to ownwith some pricep.
That is, the agent has the option to pay the pricepand obtain
the good.
Whenpis low, this approaches a full property right; whenpis
high, this approaches no right.
We can say more about the optimalpunder further regularity
conditions.

91. Illustration
By the Theorem, the best the designer can do is endow the
agent with anoption to ownwith some pricep.
That is, the agent has the option to pay the pricepand obtain
the good.
Whenpis low, this approaches a full property right; whenpis
high, this approaches no right.
We can say more about the optimalpunder further regularity
conditions.

92. The monotone case
Assumethat:
Buyer and seller virtual surpluses aremonotone:
B!()
1F!()
f!()
and
S!()+
F!()
f!()
are non-decreasing in;
Both thedesigner's and the principal's objectivefunctions
V
?
(; !)andV(; !)arenon-decreasingin the agent's type.

93. The monotone case

94. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal in the second
period will choose an optimal mechanism in which the outside-option
constraint binds for types in the interval[
?
!;
?
!].
Moreover, except for the case of a boundary solution,
V(
?
!; !) +S!(
?
!) =0;
V(
?
!; !) +B!(
?
!) =0:
For
?
!, the principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

95. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal in the second
period will choose an optimal mechanism in which the outside-option
constraint binds for types in the interval[
?
!;
?
!].
Moreover, except for the case of a boundary solution,
V(
?
!; !) +S!(
?
!) =0;
V(
?
!; !) +B!(
?
!) =0:
For
?
!, the principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

96. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal in the second
period will choose an optimal mechanism in which the outside-option
constraint binds for types in the interval[
?
!;
?
!].
Moreover, except for the case of a boundary solution,
V(
?
!; !) +S!(
?
!) =0;
V(
?
!; !) +B!(
?
!) =0:
For
?
!, the principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

97. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal in the second
period will choose an optimal mechanism in which the outside-option
constraint binds for types in the interval[
?
!;
?
!].
Moreover, except for the case of a boundary solution,
V(
?
!; !) +S!(
?
!) =0;
V(
?
!; !) +B!(
?
!) =0:
For
?
!, the principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

98. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal in the second
period will choose an optimal mechanism in which the outside-option
constraint binds for types in the interval[
?
!;
?
!].
Moreover, except for the case of a boundary solution,
V(
?
!; !) +S!(
?
!) =0;
V(
?
!; !) +B!(
?
!) =0:
For
?
!, the principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

99. Illustration: The monotone case
The optimal pricepsolves
p=
E!
h
v(!)f!(p)jp2[
?
!;
?
!]
i
E!
h
f!(p)jp2[
?
!;
?
!]
i;
assuming an interior solution.
Moreover, ifand!are independent,
p=E!
h
v(!)jp2[
?
!;
?
!]
i
:
Conditioning onp2[
?
!;
?
!]ensures that the option to own has
“bite.”
Because of the xed-point problem, the optimalpcould be
far away from the unconditional mean ofv(!).

100. Illustration: The monotone case
The optimal pricepsolves
p=
E!
h
v(!)f!(p)jp2[
?
!;
?
!]
i
E!
h
f!(p)jp2[
?
!;
?
!]
i;
assuming an interior solution.
Moreover, ifand!are independent,
p=E!
h
v(!)jp2[
?
!;
?
!]
i
:
Conditioning onp2[
?
!;
?
!]ensures that the option to own has
“bite.”
Because of the xed-point problem, the optimalpcould be
far away from the unconditional mean ofv(!).

101. Illustration: The monotone case
The optimal pricepsolves
p=
E!
h
v(!)f!(p)jp2[
?
!;
?
!]
i
E!
h
f!(p)jp2[
?
!;
?
!]
i;
assuming an interior solution.
Moreover, ifand!are independent,
p=E!
h
v(!)jp2[
?
!;
?
!]
i
:
Conditioning onp2[
?
!;
?
!]ensures that the option to own has
“bite.”
Because of the xed-point problem, the optimalpcould be
far away from the unconditional mean ofv(!).

102. Illustration: The monotone case
The optimal pricepsolves
p=
E!
h
v(!)f!(p)jp2[
?
!;
?
!]
i
E!
h
f!(p)jp2[
?
!;
?
!]
i;
assuming an interior solution.
Moreover, ifand!are independent,
p=E!
h
v(!)jp2[
?
!;
?
!]
i
:
Conditioning onp2[
?
!;
?
!]ensures that the option to own has
“bite.”
Because of the xed-point problem, the optimalpcould be
far away from the unconditional mean ofv(!).

103. Illustration: The monotone case
The optimal pricepsolves
p=
E!
h
v(!)f!(p)jp2[
?
!;
?
!]
i
E!
h
f!(p)jp2[
?
!;
?
!]
i;
assuming an interior solution.
Moreover, ifand!are independent,
p=E!
h
v(!)jp2[
?
!;
?
!]
i
:
Conditioning onp2[
?
!;
?
!]ensures that the option to own has
“bite.”
Because of the xed-point problem, the optimalpcould be
far away from the unconditional mean ofv(!).

104. Extension to contractible states
The optimal pricepsolves
p=
E!
h
v(!)f!(p)jp2[
?
!;
?
!];s
i
E!
h
f!(p)jp2[
?
!;
?
!];s
i;
assuming an interior solution.
Moreover, ifand!are independent,
p=E!
h
v(!)jp2[
?
!;
?
!];s
i
:
Ifsis a perfect signal of!, the designer can get closer to rst
best (but it is still not rst best because the principal may
sometimes buy back the agent's right).

105. Model with investment
Model with investment

106. Timeline

107. Timeline

108. Timeline

109. Timeline

110. Timeline

111. Timeline

112. Timeline

113. Timeline

114. Model
There is anagent, aprincipal, and adesigner;
At timet=0:
The designer designs a menu of rights that the agent holds.
At timet=1:
The agent decides whether to invest at (sunk) costc>0;
At timet=2:
The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

115. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

116. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

117. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

118. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

119. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

120. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

121. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

122. Model
Investment problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the investment decision is not
observed (and hence cannot be contracted upon).
In thecontractible case, the investment decision is observed
(agent's rights may be contingent on investing).
Assume: Designer wants to incentivize investment.

123. Results
Results

124. Results
Theorem
There exists an optimal menu that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal menu is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.Proof: Same as before but now the designer faces an additional
constraint that islinearin the outside-option functionR.

125. Results
Theorem
There exists an optimal menu that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal menu is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.Proof: Same as before but now the designer faces an additional
constraint that islinearin the outside-option functionR.

126. Results
Theorem
There exists an optimal menu that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal menu is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.Proof: Same as before but now the designer faces an additional
constraint that islinearin the outside-option functionR.

127. Results
Theorem
There exists an optimal menu that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal menu is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.Proof: Same as before but now the designer faces an additional
constraint that islinearin the outside-option functionR.

128. Results
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:
Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

129. Results
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:
Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

130. Results
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:
Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

131. Results
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:
Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

132. Results
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:
Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

133. Results
Corollary
Suppose that the investment cost c is sufciently high.
In the non-contractible case, there exists an optimal menu
M
?
=f(1;p);(y;p
0
)gwith y>0and p
0
=y2[;]:
In the contractible case, there exists an optimal menu
M
?
=f(1;p);(0;T)gwith with T>0.
Intuition:
When rights are contingent on investing,offering a lump-sum
paymentfor investment is effective.
When rights are not contingent, the designer can incentivize
investment only byshifting more rents to higher types.

134. Results
Corollary
Suppose that the investment cost c is sufciently high.
In the non-contractible case, there exists an optimal menu
M
?
=f(1;p);(y;p
0
)gwith y>0and p
0
=y2[;]:
In the contractible case, there exists an optimal menu
M
?
=f(1;p);(0;T)gwith with T>0.
Intuition:
When rights are contingent on investing,offering a lump-sum
paymentfor investment is effective.
When rights are not contingent, the designer can incentivize
investment only byshifting more rents to higher types.

135. Results
Corollary
Suppose that the investment cost c is sufciently high.
In the non-contractible case, there exists an optimal menu
M
?
=f(1;p);(y;p
0
)gwith y>0and p
0
=y2[;]:
In the contractible case, there exists an optimal menu
M
?
=f(1;p);(0;T)gwith with T>0.
Intuition:
When rights are contingent on investing,offering a lump-sum
paymentfor investment is effective.
When rights are not contingent, the designer can incentivize
investment only byshifting more rents to higher types.

136. Results
Corollary
Suppose that the investment cost c is sufciently high.
In the non-contractible case, there exists an optimal menu
M
?
=f(1;p);(y;p
0
)gwith y>0and p
0
=y2[;]:
In the contractible case, there exists an optimal menu
M
?
=f(1;p);(0;T)gwith with T>0.
Intuition:
When rights are contingent on investing,offering a lump-sum
paymentfor investment is effective.
When rights are not contingent, the designer can incentivize
investment only byshifting more rents to higher types.

137. Results
Corollary
Suppose that the investment cost c is sufciently high.
In the non-contractible case, there exists an optimal menu
M
?
=f(1;p);(y;p
0
)gwith y>0and p
0
=y2[;]:
In the contractible case, there exists an optimal menu
M
?
=f(1;p);(0;T)gwith with T>0.
Intuition:
When rights are contingent on investing,offering a lump-sum
paymentfor investment is effective.
When rights are not contingent, the designer can incentivize
investment only byshifting more rents to higher types.

138. Applications
Applications

139. Applications
We look at severalapplications:
1.
Details
2.
Details
3.
Details
4.
Details
5.
Details

140. Summary and future steps
Summary and future steps

141. Summary
We introduced a simple but exible framework for analyzing
optimal design of property rights.
Optimally designed rightspartially restore commitmentto
future trading mechanisms.
The optimal right often features anoption to own.

142. Summary
We introduced a simple but exible framework for analyzing
optimal design of property rights.
Optimally designed rightspartially restore commitmentto
future trading mechanisms.
The optimal right often features anoption to own.

143. Summary
We introduced a simple but exible framework for analyzing
optimal design of property rights.
Optimally designed rightspartially restore commitmentto
future trading mechanisms.
The optimal right often features anoption to own.

144. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

145. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

146. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

147. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

148. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

149. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

150. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

151. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
Whose rights take priority?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

152. A mechanism-design approach to property rights
Discussion

153. Appendix
Appendix

154. Application #1: Dynamic resource allocation
Players:
Agent: Firm
Designer: Regulator
Principal: Regulator
Agent invests in infrastructure determining value;
State!represents the value for an alternative use;
The designer maximizes a combination of efciency and
revenue:
V
?
(; !)x+
?
t= (!)x+
?
t:
The principal might put more weight on revenue:
V(; !)x+t= (!)x+t;where
?
.
Back

155. Application #1: Dynamic resource allocation
Back

156. Application #1: Dynamic resource allocation
Back

157. Application #1: Dynamic resource allocation
Back

158. Application #1: Dynamic resource allocation
Back

159. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license:
pays a lump-sum payment conditional on investing
(e.g., a ne for failing the “minimal coverage requirement”);
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
If investment isnon-contractible, the optimal license:
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
allows the holder to renew the license with some probability
conditional on paying a lower price.
Back

160. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license:
pays a lump-sum payment conditional on investing
(e.g., a ne for failing the “minimal coverage requirement”);
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
If investment isnon-contractible, the optimal license:
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
allows the holder to renew the license with some probability
conditional on paying a lower price.
Back

161. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license:
pays a lump-sum payment conditional on investing
(e.g., a ne for failing the “minimal coverage requirement”);
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
If investment isnon-contractible, the optimal license:
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
allows the holder to renew the license with some probability
conditional on paying a lower price.
Back

162. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license:
pays a lump-sum payment conditional on investing
(e.g., a ne for failing the “minimal coverage requirement”);
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
If investment isnon-contractible, the optimal license:
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
allows the holder to renew the license with some probability
conditional on paying a lower price.
Back

163. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license:
pays a lump-sum payment conditional on investing
(e.g., a ne for failing the “minimal coverage requirement”);
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
If investment isnon-contractible, the optimal license:
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
allows the holder to renew the license with some probability
conditional on paying a lower price.
Back

164. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license:
pays a lump-sum payment conditional on investing
(e.g., a ne for failing the “minimal coverage requirement”);
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
If investment isnon-contractible, the optimal license:
allows the holder to renew the license at a pre-specied
price (“renewable lease”).
allows the holder to renew the license with some probability
conditional on paying a lower price.
Back

165. Application #1: Dynamic resource allocation
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.
The optimal property right as a function of value for revenue:
=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))
Back

166. Application #1: Dynamic resource allocation
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.
The optimal property right as a function of value for revenue:
=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))
Back

167. Application #1: Dynamic resource allocation
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.
The optimal property right as a function of value for revenue:
=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))
Back

168. Application #1: Dynamic resource allocation
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.
The optimal property right as a function of value for revenue:
=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))
Back

169. Application #1: Dynamic resource allocation
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.
The optimal property right as a function of value for revenue:
=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))
Back

170. Application #2: Regulating a private rental market
Players:
Agent: Tenant
Designer: Regulator
Principal: Rental company
Agent invests in the property determining her valuefor staying.
The state!is the seller's outside option (market rental price).
The seller wants to maximize revenue:
V(; !)x+t=!x+t.
The designer wants to maximize efciency:
V
?
(; !)x+
?
t= (!)x:
Back

171. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0;andis
uniformly distributed.
Suppose that!is known (and lies in a certain range).The optimal regulation is to mandate a renewable lease with
price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:
Back

172. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0;andis
uniformly distributed.
Suppose that!is known (and lies in a certain range).The optimal regulation is to mandate a renewable lease with
price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:
Back

173. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0;andis
uniformly distributed.
Suppose that!is known (and lies in a certain range).The optimal regulation is to mandate a renewable lease with
price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:
Back

174. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0;andis
uniformly distributed.
Suppose that!is known (and lies in a certain range).The optimal regulation is to mandate a renewable lease with
price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:
Back

175. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0;andis
uniformly distributed.
Suppose that!is known (and lies in a certain range).The optimal regulation is to mandate a renewable lease with
price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:
Back

176. Application #3: Vaccine development
Players:
Agent: Pharmaceutical company
Designer: Health agency
Principal: Health agency
Company develops a vaccine; the marginal cost of production
conditional on investment isk
e
F(=k);
Allocationxis the number of units purchased by the health
agency, with 1 being the mass of the patient population;
Health agency maximizes the total value of allocation,
V
?
(; !)x=V(; !)x=!x,
where!measures the severity of the health crisis.
Assume that 1=
?
.
Back

177. Application #3: Vaccine development
Assume that investment is observable and that!is known.The optimal contract, forchigh enough, is a lump-sum payment
for the investment, plus a guaranteed sale price equal to
p
?
= min
n
!

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).
Back

178. Application #3: Vaccine development
Assume that investment is observable and that!is known.The optimal contract, forchigh enough, is a lump-sum payment
for the investment, plus a guaranteed sale price equal to
p
?
= min
n
!

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).
Back

179. Application #3: Vaccine development
Assume that investment is observable and that!is known.The optimal contract, forchigh enough, is a lump-sum payment
for the investment, plus a guaranteed sale price equal to
p
?
= min
n
!

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).
Back

180. Application #3: Vaccine development
Assume that investment is observable and that!is known.The optimal contract, forchigh enough, is a lump-sum payment
for the investment, plus a guaranteed sale price equal to
p
?
= min
n
!

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).
Back

181. Application #3: Vaccine development
If!is stochastic, then the optimal price satises
p
?
= min
(
E

!j!2[!
p
?;!p
?]


?
;

k
)
;
where[!
p
?;!p
?]is the interval of!'s for which the choice ofp
?
changes the second-period mechanism.
If the realized!is high (health crisis is severe), the principal will
offer a price better thanp
?
.
If the realized!is low (health crisis is mild), the principal will
offer a price lower thanp
?
and compensate the producer with an
additional cash payment (on top of the payment for investment).
Back

182. Application #3: Vaccine development
If!is stochastic, then the optimal price satises
p
?
= min
(
E

!j!2[!
p
?;!p
?]


?
;

k
)
;
where[!
p
?;!p
?]is the interval of!'s for which the choice ofp
?
changes the second-period mechanism.
If the realized!is high (health crisis is severe), the principal will
offer a price better thanp
?
.
If the realized!is low (health crisis is mild), the principal will
offer a price lower thanp
?
and compensate the producer with an
additional cash payment (on top of the payment for investment).
Back

183. Application #3: Vaccine development
If!is stochastic, then the optimal price satises
p
?
= min
(
E

!j!2[!
p
?;!p
?]


?
;

k
)
;
where[!
p
?;!p
?]is the interval of!'s for which the choice ofp
?
changes the second-period mechanism.
If the realized!is high (health crisis is severe), the principal will
offer a price better thanp
?
.
If the realized!is low (health crisis is mild), the principal will
offer a price lower thanp
?
and compensate the producer with an
additional cash payment (on top of the payment for investment).
Back

184. Application #4: Patent policy
Players:
Agent: Firm
Designer: Regulator
Principal: Patent agency
Firm invests in a new technology; if investment is made, the rm
can produce at constant marginal costk
~
F.
Conditional onx=1, the rm provides a monopoly quantity to
maximize prots; conditional onx=0, the rm competes in a
competitive market. Market demand isD(p) =1p.
The regulator and the patent agency maximize consumer
surplus with weight!(but attach a positive weight to revenue).
Back

185. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal policy is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.
Back

186. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal policy is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.
Back

187. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal policy is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.
Back

188. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal policy is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.
Back

189. Application #5: Supply chain contracting
Players:
Agent: Small supplier
Designer: Large producer
Principal: Large producer
Producer buys some amountxof inputs from the supplier.
The supplier must invest at timet=1 in relationship-specic
technology to produce the inputs at marginal costc .
Through the interaction with the supplier, the large rm can learn
the supplier's costs: The state!is a noisy signal of.
Producer maximizes prots having a constant marginal value 1
for each unit of the input:V!() =1 and=1:
Producer proposes a contract:V
?
!() =1 and
?
=1.
Back

190. Application #5: Supply chain contracting
Suppose investment by the small supplier isnot contractible.
The producer will in general commit to atwo-price scheme,
committing to buy up toyunits at some pricep
H, and any
number of units at some lower pricep
L.
If investment by the small supplier iscontractible, assuming the
cost of investment is high enough, the producer will offer an
upfront paymentfor setting up production and aguaranteed
purchase pricefor any number of units.
Back

191. Application #5: Supply chain contracting
Suppose investment by the small supplier isnot contractible.
The producer will in general commit to atwo-price scheme,
committing to buy up toyunits at some pricep
H, and any
number of units at some lower pricep
L.
If investment by the small supplier iscontractible, assuming the
cost of investment is high enough, the producer will offer an
upfront paymentfor setting up production and aguaranteed
purchase pricefor any number of units.
Back

192. Application #5: Supply chain contracting
Suppose investment by the small supplier isnot contractible.
The producer will in general commit to atwo-price scheme,
committing to buy up toyunits at some pricep
H, and any
number of units at some lower pricep
L.
If investment by the small supplier iscontractible, assuming the
cost of investment is high enough, the producer will offer an
upfront paymentfor setting up production and aguaranteed
purchase pricefor any number of units.
Back

193. Ironing Jullien (2000)
Back

194. Ironing Jullien (2000)
Back

195. Ironing Jullien (2000)
Back

196. Ironing Jullien (2000)
Back

197. Ironing Jullien (2000)
Back

198. Ironing Jullien (2000)
Back
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

199. Ironing Jullien (2000)
Back
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

200. Ironing Jullien (2000)
Back
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

201. Ironing Jullien (2000)
Back
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

202. Ironing Jullien (2000)
Back
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

203. Ironing Jullien (2000)
Back
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)

204. Ironing Jullien (2000)
Back
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour
R
Dworczak
R
Kominers (2023)