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Material Type: Paper; Class: GRECO-ROMAN ARCHTCT; Subject: Classics; University: University of California - Los Angeles; Term: Unknown 2004;
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Robust Mechanism Design
Abstract The mechanism design literature assumes too much common knowledge of the en- vironment among the players and planner. We relax this assumption by studying im- plementation on richer type spaces. We ask when ex post implementation is equivalent to interim (or Bayesian) imple- mentation for all possible type spaces. The equivalence holds in the case of separable environments; examples of separable environments arise (1) when the planner is imple- menting a social choice function (not correspondence); and (2) in a quasilinear envi- ronment with no restrictions on transfers. The equivalence fails in general, including in some quasilinear environments with budget balance. In private value environments, ex post implementation is equivalent to dominant strategies implementation. The private value versions of our results o§er new insights into the relation between dominant strategy implementation and Bayesian implemen- tation. Keywords: Mechanism Design, Common Knowledge, Universal Type Space, Interim Equilibrium, Ex-Post Equilibrium, Dominant Strategies. Jel Classification: C79, D
This research is supported by NSF Grant #SES-0095321. We would like to thank the co-editor, three anonymous referees and seminar participants at many institutions for helpful comments. We thank Bob Evans for pointing out errors in earlier examples and Sandeep Baliga, Matt Jackson, Jon Levin, Bart Lipman, Eric Maskin, Zvika Neeman, Andrew Postlewaite, Ilya Segal and Tomas Sjˆstrˆm for valuable discussions. yDepartment of Economics, Yale University, 28 Hillhouse Avenue, New Haven, CT 06511, [email protected]. zDepartment of Economics, Yale University, 30 Hillhouse Avenue, New Haven, CT 06511, [email protected].
suited to this task. In fact, Harsanyiís work was intended to address the then prevailing criticism of game theory that the very description of a game embodied common knowledge assumptions that could never prevail in practise. Harsanyi argued that by allowing an agentís type to include his beliefs about the strategic environment, his beliefs about other agentsí beliefs, and so on, any environment of incomplete information could be captured by a type space. With this su¢ ciently large type space (including all possible beliefs and higher order beliefs), it is true (tautologically) that there is common knowledge among the agents of each agentís set of possible types and each typeís beliefs over the types of other agents. However, as a practical matter, applied economic analysis tends to assume much smaller type spaces than the universal type space, and yet maintain the assumption that there is common knowledge among the agents of each agentís type space and each typeís beliefs over the types of other agents. In the small type space case, this is a very substantive restriction. There has been remarkably little work since Harsanyi checking whether analysis of incomplete information games in economics is robust to the implicit common knowledge assumptions built into small type spaces.^3 We will investigate the importance of these implicit common knowledge assumptions in the context of mechanism design.^4 Formally, we Öx a payo§ environment, specifying a set of payo§ types for each agent, a set of outcomes, utility functions for each agent and a social choice correspondence (SCC) mapping payo§ type proÖles into sets of acceptable outcomes. The planner (partially) im- plements^5 the social choice correspondence if there exists a mechanism and an equilibrium strategy proÖle of that mechanism such that equilibrium outcomes for every payo§ type proÖle are acceptable according to the SCC.^6 This is sometimes referred to as Bayesian implementation, but since we do not have a common prior, we will call it interim imple- mentation. While holding Öxed this environment, we can construct many type spaces, where an agentís type speciÖes both his payo§ type and his belief about other agentsítypes. Crucially, there may be many types of an agent with the same payo§ type. The larger the type space, the harder it will be to implement the social choice correspondence, and so the more ìrobustî the resulting mechanism will be. The smallest type space we can work with is the ìpayo§ type space,î where we set the possible types of each agent equal to the set (^3) Battigalli and Siniscalchi (2003), Morris and Shin (2003). (^4) Neeman (2001) argued that small type space assumptions are especially important in the full surplus extraction results of Cremer and McLean (1985). (^5) "Partial implementation" is sometimes called "truthful implementation" or "incentive compatible im- plementation." Since we look exclusively at partial implementation in this paper, we will write "implement" instead of "partially implement". (^6) In a companion paper, Bergemann and Morris (2004), we use the framework of this paper to look at full implementation, i.e., requiring that every equilibrium delivers an outcome consistent with the social choice correspondence.
of payo§ types, and assume a common knowledge prior over this type space. This is the usual exercise performed in the mechanism design literature. The largest type space we can work with is the union of all possible type spaces that could have arisen from the payo§ environment. This is equivalent to working with a ìuniversal type space,î in the sense of Mertens and Zamir (1985). There are many type spaces in between the payo§ type space and the universal type space that are also interesting to study. For example, we can look at all payo§ type spaces (so that the agents have common knowledge of a prior over payo§ types but the mechanism designer does not); and we can look at type spaces where the common prior assumption holds. In the face of a planner who does not know about agentsíbeliefs about othersípayo§ types, a recent literature has looked at mechanisms that implement the SCC in ex post equilibrium (see references in footnote 9). This requires that in a payo§ type direct mech- anism - where each agent is asked to report his payo§ type - each agent has an incentive to tell the truth if he expects others to tell the truth, whatever their types turn out to be. In the special case of private values, ex post implementation is equivalent to dominant strategies implementation. If an SCC is ex post implementable, then it is clearly interim implementable on every type space, since the payo§ type direct mechanism can be used to implement the SCC. The converse is not always true. In Examples 1 and 2, ex post implementation is impossible. Nonetheless, interim implementation is possible on every type space. The gap arises because the planner may have the equilibrium outcome depend on the agentsíhigher order belief types, as well as their realized payo§ type. The planner has no intrinsic interest in conditioning on non-payo§-relevant aspects of agentsítypes, but he is able to introduce slack in incentive constraints by doing so. The main question we address in this paper is when the converse is true. A payo§ en- vironment is separable if the outcome space has a common component and a private value component for each agent. Each agent cares only about the common component and his own private component. The social choice correspondence picks a unique element from the common component and has a product structure over all components. In separable envi- ronments, interim implementation on all common prior payo§ type spaces implies ex post implementation.^7 Whenever the social choice correspondence is a function, the environment has a separable representation (since we can make private value components degenerate). The other leading example of a separable environment is the problem of choosing an al- location when arbitrary transfers are allowed and agents have quasi-linear utility. If the allocation choice is a function but the planner does not care about the level and distribu- (^7) This result extends to all common prior full support type spaces in the quasilinear case and when the environment is compact.
We consider a Önite set of agents 1 ; 2 ; :::; I. Agent iís payo§ type is i 2 i, where i is a Önite set. We write 2 = 1 ::: I. There is a set of outcomes Y. Each agent has utility function ui : Y ! R. A social correspondence is a mapping F : ! 2 Y^ n;. If the true payo§ type proÖle is , the planner would like the outcome to be an element of F (). An important special case - studied in some of our examples and results - is a quasi-linear environment where the set of outcomes Y has the product structure Y = Y 0 Y 1 YI , where Y 1 = Y 2 = :: = YI = R, and a utility function:
ui (y; ) = ui (y 0 ; y 1 ; :::; yI ; ) , vi (y 0 ; ) + yi
which is linear in yi for every agent i. The planner is concerned only about choosing an "allocation" y 0 2 Y 0 and does not care about transfers. Thus there is a function f 0 : ! Y 0 and F () = f(y 0 ; y 1 ; :::; yI ) 2 Y : y 0 = f 0 ()g. Throughout the paper, this environment is Öxed and informally understood to be com- mon knowledge. We allow for interdependent types - one agentís payo§ from a given outcome depends on other agentsípayo§ types. The payo§ type proÖle is understood to contain all information that is relevant to whether the planner achieves his objective or not. For exam- ple, we do not allow the planner to trade o§ what happens in one state with what happens in another state. For the latter reason, this setup is somewhat restrictive. However, it in- corporates many classic problems such as the e¢ cient allocation of an object or the e¢ cient provision of a public good.
While maintaining that the above payo§ environment is common knowledge, we want to allow for agents to have all possible beliefs and higher order beliefs about other agentsí types. A áexible framework for modelling such beliefs and higher order beliefs are type spaces. A type space is a collection T =
Ti; bi; bi
i=1 : Agent iís type is ti 2 Ti. A type of agent i must include a description of his payo§ type. Thus there is a function bi : Ti! i,
with bi (ti) being agent iís payo§ type when his type is ti. A type of agent i must also include a description of his beliefs about the types of the other agent. Write (Z) for the space of probability measures on the Borel Öeld of a measurable space Z. The belief of type ti of agent i is a function bi : Ti! (T i) ,
with bi (ti) being agent iís belief type when his type is ti. Thus bi (ti) [E] is the probability that type ti of agent i assigns to other agentsítypes, t i, being an element of a measurable set E T i. In the special case where each Tj is Önite, we will abuse notation slightly by writing bi (ti) [t i] for the probability that type ti of agent i assigns to other agents having types t i.
Fix a payo§ environment and a type space T. A mechanism speciÖes a message set for each agent and a mapping from message proÖles to outcomes. Social choice correspondence F is interim implementable if there exists a mechanism and an interim (or Bayesian) equilibrium of that mechanism such that outcomes are consistent with F. However, by the revelation principle, we can restrict attention to truth-telling equilibria of direct mechanisms.^8 A direct mechanism is a function f : T! Y.
DeÖnition 1 A direct mechanism f : T! Y is interim incentive compatible on type space T if Z
t i 2 T i
ui
f (ti; t i) ; b (ti; t i)
dbi (ti)
t i 2 T i
ui
f
t^0 i; t i
; b (ti; t i)
dbi (ti)
for all i, t 2 T and t^0 i 2 Ti.
The notion of interim incentive compatibility is often referred to as Bayesian incentive compatibility. We use the former terminology as there need not be a common prior on the type space.
DeÖnition 2 A direct mechanism f : T! Y on T achieves F if
f (t) 2 F
b (t)
for all t 2 T. (^8) See Myerson (1991), Chapter 6.
We are interested in characterizing interim incentive compatibility on di§erent type spaces. We Örst introduce some key properties of type spaces. A type space T is a payo§ type space if each Ti = i and each bi is the identity map. Type space T is Önite if each Ti is Önite. Finite type space T has full support if bi (ti) [t i] > 0 for all i and t. Finite type space T satisÖes the common prior assumption (with prior p) if there exists p 2 (T ) such that X t i 2 T i
p (ti; t i) > 0 for all i and ti
and bi (ti) [t i] = Pp^ (ti; t i) t^0 i 2 T i
p
ti; t^0 i
The standard approach in the mechanism design literature is to restrict attention to a common prior payo§ type space (perhaps with full support). Thus it is assumed that there is common knowledge among the agents of a common prior over the payo§ types. A payo§ type space can be thought of the smallest type space embedding the payo§ environment described above. Restricting attention to a full support, common prior, payo§ type space is with loss of generality. We want to relax the implicit common knowledge assumptions embodied in those restrictions by asking the following progressively tougher questions about interim implementability:
Is F interim implementable on all full support common prior payo§ type spaces?
Is F interim implementable on all common prior payo§ type spaces?
Is F interim implementable on all common prior type spaces?
Is F interim implementable on all type spaces?
By requiring that F be interim implementable on all type spaces, we are asking for a mechanism that can implement F with no common knowledge assumptions beyond those in the speciÖcation of the payo§ environment. If we constructed a universal type space for the payo§ environment, that universal type space would be an example of a type space and thus interim implementability on all type spaces would imply interim implementability on the universal type space. We discuss the relation between our approach and the universal type space in more detail in Section 6. We will see that relaxing common knowledge assumptions makes a di§erence. In par- ticular, we will show that while the common prior assumption is not important and the full
support assumption does not play a big role,^10 the payo§ type space restriction is important. In example 3 in the next section, it is possible to interim implement on any payo§ type space (with or without the common prior) but not all type spaces. We are especially interested in the relation between the ex post implementability of F and interim implementability. In Sections 4 and 5, we provide su¢ cient conditions for ex post implementability to be equivalent to interim implementability on all type spaces. But Examples 1 and 2 in the next section show that it is possible to Önd social choice correspondences that are interim implementable on any type space but not ex post implementable.
This section presents three examples illustrating the relationship between interim imple- mentation on di§erent type spaces and ex post implementation. The Örst two examples exhibit social choice correspondences that are interim imple- mentable on all type spaces, but are not ex post implementable. The Örst example is very simple, but relies on (i) a restriction to deterministic allocations, (ii) a social choice cor- respondence that depends on only one agentís payo§ type; and (iii) interdependent types. In the second example, we show how to dispense with all three features. Since this second example has private values, we thus have an example where dominant strategies implemen- tation is impossible but interim implementation is possible on any type space. The third example exhibits a social choice correspondence that is interim implementable on all payo§ type spaces (with or without the common prior) but is not interim imple- mentable on all type spaces. The social choice correspondence represents e¢ cient alloca- tions in a quasi-linear environment with a balanced budget requirement. As such it also illustrates some of the results presented in later in Section 5 on social choice problems with a balanced budget.
Example 1 There are two agents. Each agent has two possible types: 1 =
1 ; ^01 and 2 =
2 ; ^02. There are three possible allocations: Y = fa; b; cg. The payo§s of the two agents are given by the following tables (each box describes agent 1ís payo§, then agent 2ís 1 0 (^) However, di§erent type space assumptions will be important for di§erent questions. The full support assumption is crucial when we look at full implementation (see Bergemann and Morris (2004)) and the common prior assumption is important when we look at revenue maximization (see Bergemann, Morris and Segal (2004)).
di§erent outcomes depending on agent 2 ís type. Now instead of having agent 1 ís utility depend on agent 2 ís type, it can depend on the plannerís reÖned choice.
Example 2 There are two agents. Agent 1 has three possible types, 1 =
and agent 2 has two possible types, 2 =
2 ; ^02. There are eight possible pure allo- cations, fa; b; c; d; a^0 ; b^0 ; c^0 ; d^0 g, and lotteries are allowed, so Y = (fa; b; c; d; a^0 ; b^0 ; c^0 ; d^0 g). The private value payo§s of agent 1 are given by the following table:
u 1 a b c d a^0 b^0 c^0 d^0 1 1 1 12 1 1 1 (^1 ) ^01 0 0 1 0 0 0 1 ^001 0 0 0 1 0 0 0
The private value payo§s of agent 2 are given by the following table:
u 2 a b c d a^0 b^0 c^0 d^0 2 0 1 0 0 0 1 1 1 ^02 1 0 1 1 1 0 0 0
The social choice correspondence F is described by the following table.^11
2 ^02 1 fa; bg fa^0 ; b^0 g ^01 fcg fc^0 g ^001 fdg fd^0 g
We now show - by contradiction - that this correspondence is not ex post implementable. Let q be the probability that a is chosen at proÖle ( 1 ; 2 ) and let q^0 be the probability that a^0 is chosen at proÖle
. In order for type 1 to have an incentive to tell the truth (and not report himself to be type ^01 ) when he is sure that agent 2 is type 2 , we must have
q (1 q)
2 ,^ q^ ^
In order for type 1 to have an incentive to tell the truth (and not report himself to be type ^001 ) when he is sure that agent 2 is type ^02 , we must have
q^0 +
1 q^0
2 ,^ q
1 1 (^) The SCC F in this example is not ex post Pareto e¢ cient at ( 1 ; 2 ) and ( 1 ; (^02) ), as b (^0) and a, respectively, Pareto dominate b and a^0 , respectively. We choose this example for the simplicity of its payo§s. Yet, we have constructed examples with the same number of agents, states and allocations such that the SCC F is ex ante Pareto e¢ cient and interim implementable on all type spaces, but not ex post, and a fortiori, not dominant strategy implementable.
But in order for agent 2 to have an incentive to tell the truth when he is type 2 and he is sure that agent 1 is type 1 , we must have
1 q 1 q^0 ;
thus q^0 q. (3)
However, (1), (2) and (3) generate a contradiction, so ex post implementation is not possible. But it is straightforward to implement on any interim type space. Consider the following indirect mechanism for any arbitrary type space where individual 1 chooses a message m 1 2 m^11 ; m^21 ; m^31 ; m^41 and individual 2 chooses a message m 2 2
m^12 ; m^22 and let outcomes be chosen as follows: m^12 m^22 m^11 a a^0 m^21 b b^0 m^31 c c^0 m^41 d d^0
There is always an equilibrium where type 1 of agent 1 sends message m^11 if he believes agent 2 is type 2 with probability at least 12 and message m^21 if he believes agent 2 is type 2 with probability less than 12 ; type ^01 always sends message m^31 ; and type ^001 always sends message m^41. Type 2 of agent 2 sends message m^12 and type ^02 sends message m^22 , and this strategy is a dominant strategy for agent 2. This private values example has the feature that dominant strategies implementation is impossible but interim implementation is possible on any type space, and seems to be the Örst example in the literature noting this possibility.^12 As we will see in the next section, a necessary feature of the example is that we have a social choice correspondence (not function) that we are trying to implement. In the example, it was further key that there were aspects of the allocation that the planner did not care about but the agents did. In the example, this may look a little contrived but note that this a natural feature of quasi-linear environments where the planner wants to maximize the total welfare of agents. We will next present a quasi-linear utility example that exploits this feature. 1 2 (^) It is often noted that in public good problems with budget balance, dominant strategies implementation is impossible while Bayesian implementation is possible. However, the positive Bayesian implementation results (díAspremont and Gerard-Varet (1979) and díAspremont, Cremer and Gerard-Varet (1995, 2002)) hold only for "generic" priors on a Öxed type space, not for all type spaces in our sense. They provide examples showing that Bayesian implementation fails for some type spaces.
It is straightforward to establish that ex post implementation with balanced transfers is not feasible. Writing fi () for the transfer received by i at payo§ type proÖle , we have the following ex post incentive constraints for agent 1 :
v 1 (f 0 ( 1 ; 2 ) ; ( 1 ; 2 )) + f 1 ( 1 ; 2 ) v 1
f 0
v 1
f 0
v 1
f 0
and
v 1
f 0
v 1
f 0
v 1
f 0
v 1
f 0
Inserting the gross utilities v 1 (; ), we can write the above set of inequalities as follows
f 1 ( 1 ; 2 ) f 1
f 1
and f 1
f 1
f 1
Next we consider the ex post incentive constraints for agent 2 , at 1 and ^001 , respectively. Here the social choice mapping prescribes allocations constant in the reported type proÖle of agent 2 and ex post incentive compatibility hence requires constant transfers as well, or f 2 ( 1 ; 2 ) = f 2
and f 2
= f 2
. Using the balanced budget requirement by writing f 2 () = f 1 (), we thus obtain
f 1 ( 1 ; 2 ) = f 1
and f 1
= f 1
which leads to a contradiction with the inequalities (5) and (6). Despite the failure of ex post implementation, we now show that we can satisfy the interim incentive compatibility conditions for every prior on the payo§ type space. The sole determinant of the appropriate transfers is the belief of agent 1 with payo§ type ^01. If type ^01 assigns probability at least 12 to agent 2 being of payo§ type 2 , then the following transfers to agent 1 (and corresponding balanced budget transfers for agent 2 ) are interim incentive compatible: f 1 ( 1 ; 2 ) = 0 f 1
f 1
= 0 f 1
f 1
= 1 f 1
Conversely, if type ^01 assigns probability less than 12 to the other agent being of payo§ type 2 , then the following transfers to agent 1 are interim incentive compatible:
f 1 ( 1 ; 2 ) = 1 f 1
f 1
= 1 f 1
f 1
= 0 f 1
By symmetry of the payo§s, it will su¢ ce to verify the incentive compatibility conditions for the Örst case. We Örst observe that all the ex post incentive constraints hold except for agent 1 at type proÖle ^01 ^02 , where he has a proÖtable deviation by misreporting himself to be of type 1. Suppose then that type ^01 assigns probability p to the other agent being type 2. His expected payo§ to truth-telling is
p (0 + 0) + (1 p) (0 1) = (1 p)
while his expected payo§ to mis-reporting type 1 is
p ( 4 + 0) + (1 p) (1 + 0) = 1 5 p;
and his expected payo§ to mis-reporting type ^001 is given by
p (1 1) + (1 p) ( 4 1) = 5 (1 p) :
Thus truth-telling is optimal as long as
(1 p) 1 5 p , p
The second set of transfers, described in (8), o§er interim incentive compatibility for agent 1 provided that p 23. As either of the above transfer schemes satisÖes the ex post incentive constraints of agent 2 , it follows for every belief p by type ^01 , we can Önd interim incentive compatible transfers and hence F is interim implementable for all payo§ type spaces. However, on richer type spaces than the payo§ type space, there may be many types with payo§ type ^01 , some of whom are sure that the other agent is type 2 while others are sure that he is type ^02. That is the idea behind the following example of a ìcomplete informationîtype space where F cannot be interim implemented. We consider the following
tk 1 +1(modulo 6 ) imply (for k = 1; 2 ; ::; 6 respectively):
0 + y 11 0 + y 21 0 + y 22 1 + y 32 0 + y 33 0 + y 43 (10) 0 + y 44 0 + y 54 0 + y 55 1 + y 65 0 + y 66 0 + y 16
The incentive constraints corresponding to types tl 2 mis-reporting to be type tl 2 1 imply, using the balanced budget to write the transfers to agent 2 as the negatives of agent 1 , (for l = 1; 2 ; ::; 6 respectively):
2 y 11 2 y 16 2 y 22 2 y 21 2 y 33 2 y 32 (11) 2 y 44 2 y 43 2 y 55 2 y 54 2 y 66 2 y 65
The inequalities (10) and (11) have a very simply structure. With very few exceptions, the payo§s appearing on the lhs and rhs of the inequalities are identical and only the transfers di§er. These inequalities are generated either by true and misreported types which induce only di§erent transfer decisions but identical allocational decisions. The exceptions are the second and Öfth inequality of agent 1 , where a misreported type also leads to a di§erent allocational decision. Re-arranging the inequalities, we obtain
0 y 21 y 11 ; 0 y 11 y 16 1 y 32 y 22 ; 0 y 22 y 21 0 y 43 y 33 ; 0 y 33 y 32 0 y 54 y 44 ; 0 y 44 y 43 1 y 65 y 55 ; 0 y 55 y 54 0 y 16 y 66 ; 0 y 66 y 65
When we sum these twelve constraints, the transfers on the right hand side of the inequal- ities cancel out and we are left with the desired contradiction for any arbitrary choice of probabilities, namely 2 0. The transfers cancelled out because the set of incentive constraints for agent 1 and agent 2 formed jointly a cycle through the type space.
We now present general results about the relationship between ex post implementability and interim implementability on di§erent type spaces. The Örst result is an immediate implication from the deÖnition of ex post equilibrium.
Proposition 1 If F is ex post implementable, then F is interim implementable on any type space.
PROOF: If F is ex post implementable, then by hypothesis there exists f ^ : ! Y with f ^ () 2 F () for all , such that for all i, all and all ^0 i:
ui (f ^ () ; ) ui
f ^
^0 i; i
Consider then an arbitrary type space T and the direct mechanism f : T! Y with f (t) = f
b (t)
. Incentive compatibility now requires
ti 2 arg max t^0 i 2 Ti
t i 2 T i
ui
f
t^0 i; t i
bi (ti) ; b i (t i)
dbi (ti)
= arg max t^0 i 2 Ti
t i 2 T i
ui
f ^
bi^ t^0 i^ ^ ; b i (t i)
bi (ti) ; b i (t i)
dbi (ti).
This requires that
bi (ti) = arg max i 2 i
t i 2 T i
ui
f ^
i; b i (t i)
bi (ti) ; b i (t i)
dbi (ti)
= arg max i 2 i
i 2 i
ft i:b i(t i)= ig
dbi (ti)
ui
f ^ (i; i) ;
bi (ti) ; i
But by hypothesis of ex post implementability, truthtelling is a best response for every possible proÖle i, and thus it remains a best response for arbitrary expectations over i.
The converse does not always hold, as shown by Examples 1 and 2 in the previous section. But we can identify important classes of problems for which the equivalence can be established.
A social choice environment is separable if the outcome space has a common component and a private value component for each agent. Each agent cares only about the common compo- nent and his own private value component. The social choice correspondence picks a unique element from the common component, and has a product structure over all components.
