May 20, 2014
Some Motivation Previously we considered the problem of matching workers with firms We considered some different institutions for tackling the incentive problem arising from asymmetric information What if we could design the institution? What is possible? Formally an institution is a game.
Social Choice Problems A set of individuals i = 1,..., N. Y : the set of alternatives, Θ i : the set of possible types for i N ˆv i (y, θ): utility index of i if the type profile is θ j Θ j. For today and most of this class, we will specialize to private values: ˆv i (y, θ) = ˆv i (y, θ i )
Social Choice Functions Definition A social choice function is a mapping f : Θ Y which describes the desired outcome as a function of the agents types.
Examples Single object/single buyer (tioli) Auction setting. etc. Examples of social choice functions.
Game Forms Consider any extensive-form where The players are the individuals i = 1,..., N Attached to terminal nodes are elements of Y. We can then write g(σ) for the element of Y that is attached to the terminal node reached by σ. And consider the Bayesian extensive-form game in which the players are first privately informed of their types and then they play out the extensive form.
Dominant Strategies Definition Given an extensive-form Γ, suppose that for each player i there is a strategy profile σ ˆv i (g(σ(θ)), θ i ) ˆv i (g(ˆσ i (θ i ), σ i (θ i )), θ i ) for all θ, and for all ˆσ i (θ i ). Then we say that σ is an (ex post) dominant-strategy solution of Γ. It is standard in mechanism design to use this definition of dominant strategies.
Implementation Definition If there is an extensive-form Γ with a dominant strategy solution σ such that f (θ) = g(σ(θ)), then we say that the social choice function f is implemented in dominant strategies by Γ. We refer to Γ as the implementing mechanism. And we say that the social choice function f is implementable in dominant strategies.
Incentive Compatibility Proposition Suppose that the social choice function f is implementable in dominant strategies. Then the dominant strategy incentive compatibility constraints are satisfied: ˆv i (f (θ), θ i ) ˆv i (f ( ˆθ i, θ i ), θ i ) for all θ i, θ i, ˆθ i.
Incentive Compatibility Definition If a social choice function satisfies the dominant-strategy incentive-compatibility constraints we say that it is dominant strategy incentive compatible, or DSIC.
Examples Continued Check incentive compatibility of the social choice functions from the examples.
Direct Revelation Mechanisms Consider the simplest possible game form to implement a social choice function f. Simultaneous moves. Each player s action set A i is simply Θ i Essentially, the players are simultaneously announcing types. Of course nothing stops them from lying. The truthful strategy for i is σ i (θ i ) = θ i for all θ i. g(θ) = f (θ). Note that this mechanism is uniquely defined for a given f. We call it the direct-revelation mechanism associated with the social choice function f.
Incentive Compatible Direct Revelation Mechanisms When will the truthful strategies be a dominant strategy solution of a direct revelation mechanism? Proposition Given a social choice function f, if the dominant strategy incentive compatibility constraints are satisfied then the truthful strategies are a dominant strategy solution of the direct-revelation mechanism associated with f.
The Revelation Principle The Revelation Principle says that in order to implement a social choice function f, we don t need to consider games that are any more complicated than direct-revelation mechanisms. Proposition A social choice function is implementable in dominant strategies if and only if it is DSIC
Monetary Transfers and Quasi-Linear Utilities Today we will examine a setup that is typical in economic applications. Monetary transfers: Y = X R N, so y = (x, t) where x X is an alternative that affects all agents. t = (t 1,..., t N ) is a monetary transfer scheme where t i is the payment made by agent i. Quasi-linear utilities: ˆv i (y, θ i ) = v i (x, θ i ) t i For example, the single object/single buyer example and the auction example.
DSIC Mechanisms For This Setting A social choice function has two components α : Θ X is allocation rule, τ : Θ R N is the set of transfers. In this setting, DSIC means that i N, θ i Θ i, v i (α(θ i, θ i ), θ i ) τ i (θ i, θ i ) v i (α( ˆθ i, θ i ), θ i ) τ i ( ˆθ i, θ i ), for all ˆθ i Θ i. The pair (α, τ) is also referred to as a mechanism.
Efficient Mechanisms Definition An efficient mechanism is a mechanism Γ = (α, τ), where the allocation rule is such that α(θ) arg max x X N j=1 v j (x, θ j ).
The Efficient Allocation for the Auction Example
The Vickrey-Clarke-Groves Mechanism The VCG mechanism (Vickrey-Clarke-Groves mechanism) is a mechanism Γ = (α, τ), where α is the efficient allocation rule and the transfer rule τ is defined as follows. τ i (θ) = i =j v j (α( θ i, θ i ), θ j ) v j (α(θ), θ j ) i =j where θ = ( θ 1,..., θ n ) is a pre-specified profile of default types and
Important The VCG mechanism is a direct-revelation game. All of the θ s in the formula are the announced θ s of the players, as opposed to their actual θ s. The rules of the mechanism can never depend on the actual θ s of the players.
VCG Mechanism For The Auction Example
VCG is DSIC Proposition For any profile of default types the VCG mechanism is dominant strategy incentive compatible.
Further Specialization Linear values: Θ i = [0, 1]. v i (x, θ i ) = θ i v i (x).
Example: Bilateral Trade Two individuals, buyer and seller. Seller possesses an indivisible object which the buyer is potentially interested in. The alternatives are X = {trade, notrade}. Both have zero value from no trade: v b (notrade) = v s (notrade) = 0. Buyer has a value θ b for the good and seller has cost θ s for selling the good: v b (x) = x v s (x) = x so that the buyer s utility from trading with probability x and paying t b is θ b v b (x) t b = xθ b t b and the seller s utility from trading with probabilty x and receiving t s is θ s v s (x) + t s = t s xθ s
Efficient Trade The efficient allocation rule is α (θ) = { trade notrade if θ b > θ s otherwise We know that the VCG mechanism is a DSIC efficient mechanism. Let s calculate the transfers τ(θ). We will pick the default types to be θ b = 0, θ s = 1. When there is no trade, τ b (θ) = τ s (θ) = 0. When there is trade, i.e. when θ b > θ s, τb VCG (θ) = θ s (buyer pays seller s cost), (θ) = θ b (and seller receives buyers value). τ VCG s There is a deficit since θ b > θ s.
A Convenient Characterization of DSIC Call U i (θ) the ex-post (or indirect) utility of the mechanism. DSIC can be expressed using U i : U i (θ) = θ i v i (α(θ)) τ(θ). U i (θ) θ i v i (α( ˆθ i, θ i )) τ( ˆθ i, θ i ) = ˆθ i v i (α( ˆθ i, θ i )) + (θ i ˆθ i )v i (α( ˆθ i, θ i )) τ( ˆθ i, θ i ) = U i ( ˆθ i, θ i ) + (θ i ˆθ i )v i (α( ˆθ i, θ i )) This is a way of expressing the constraint that θ i does not want to misreport his type to be ˆθ i.
A Convenient Characterization of DSIC Similarly, DSIC means that type ˆθ i does not want to misreport his type to be θ i : U i ( ˆθ i, θ i ) U i (θ) + ( ˆθ i θ i )v i (α(θ)). From these two conditions, assuming WLOG that θ i > ˆθ i it follows that θ i Θ i v i (α(θ)) U i (θ) U i ( ˆθ i, θ i ) θ i ˆθ i v i (α( ˆθ i, θ i )).
Monotonicity Lemma If (α, τ) is DSIC, then θ i > ˆθ i v i (α(θ)) v i (α( ˆθ i, θ i )). That is, v i (α(, θ i )) is an increasing function, θ i.
Smoothness Since a monotonic function is continuous almost everywhere, lim v i (α( ˆθ i, θ i )) = v i (α(θ)), ˆθ i θ i almost everywhere, therefore recalling that we obtain v i (α(θ)) U i (θ) U i ( ˆθ i, θ i ) θ i ˆθ i v i (α( ˆθ i, θ i )). U i (θ i, θ i ) θ i U i (θ) U i ( ˆθ i, θ i ) = lim = v i (α(θ)). ˆθ i θ i θ i ˆθ i Thus U i is differentiable almost everywhere.
Payoff Equivalence Monotonicity implies that this derivative is increasing, thus U i is a convex differentiable function. A convex differentiable function is the integral of its derivative: Lemma The indirect utility function from a DSIC mechanism satisfies θ i U i ( θ i, θ i ) = U i (0, θ i ) + v i (α(s, θ i ))ds, θ i. 0 This is called payoff equivalence (also envelope condition or Mirrlees condition)
Payoff Equivalence This means that any two DSIC mechanisms implementing the same allocation rule α give the same indirect utility function up to a family of constants (the numbers U i (0, θ i ).) In particular, for any pair of types θ i, θ i of i and any profile θ i of types of the other agents, U i (θ i, θ i ) U i (θ i, θ i ) = Û i (θ i, θ i ) Û i (θ i, θ i ) where U and Û represent the indirect utility functions of the two mechanisms.
Revenue Equivalence Since U i (θ) = θ i v i (α(θ)) τ i (θ) another way of saying this is that the transfer rule is unique up to a family of constants. That is τ i (θ i, θ i ) τ i (θ i, θ i ) = ˆτ i (θ i, θ i ) ˆτ i (θ i, θ i ) This is the revenue equivalence theorem.
Applying Payoff Equivalence To The Bilateral Trade Example Let τb VCG ( ) and τs VCG ( ) be the transfer functions in the VCG mechanism for the bilateral trade problem. By payoff equivalence, any DSIC efficient mechanism has a transfer rule ˆτ satisfying ˆτ b (θ) ˆτ b (0, θ s ) = τb VCG (θ) τb VCG (0, θ s ) ˆτ s (θ) ˆτ s (θ b, 1) = τ VCG s (θ) τ VCG s (θ b, 1). Substituting the expressions for the VCG transfers in this problem and rearranging, we obtain ˆτ b (θ) = θ s + ˆτ s (0, θ s ) ˆτ s (θ) = θ b + ˆτ s (θ b, 1).
Individual Rationality Typically we assume that individuals cannot be compelled to participate in the mechanism. If we normalize their reservation utility to be zero, then we can express this individual rationality constraint: U i (θ) 0 for all θ. This is an ex post individual rationality constraint as it is required to hold for all θ which is in the spirit of DSIC.
Individual Rationality in The Bilateral Trade Example If the mechanism is ex-post individually rational then ˆτ b (0, θ s ) 0 ˆτ s (θ b, 1) 0 because types θ b = 0 and θ s = 1 will have a zero value (regardless of the mechanism.)
Budget Surpluses and Deficits Definition The (ex-post) budget surplus of a mechanism at a type profile θ is given by S(θ) = τ i (θ) i If S(θ) 0 for all θ then we say that the mechanism never runs a deficit. If S(θ) = 0 for all θ then we say that the transfer rule and the associated mechanism are ex post budget balanced.
Budget Surpluses in The Bilateral Trade Example Consider any efficient DSIC mechanism in the bilateral trade example. The ex-post budget surplus at a profile θ where α(θ) = trade will be equal to τb VCG (θ) + τs VCG (θ) + ˆτ(0, θ s ) + ˆτ(θ b, 0). and since ˆτ(0, θ s ) 0 and ˆτ(θ b, 0) 0, any efficient DSIC mechanism has a smaller ex-post budget surplus than the VCG mechanism (with default type profile θ = (0, 1).)
Budget Balance in the Bilateral Trade Example Recall that the VCG mechanism runs a deficit at every profile where θ b > θ s. Hence Proposition There does not exist an efficient, DSIC, ex-post IR, budget-balancing mechanism for the bilateral trade problem.