1. Incentives and Mechanism Design Introduction ; An important features of any setting in which collective decisions must be made is that individual actual preferences are not publicly observable. As a result, in one way or another, individuals must be relied upon to reveal this information. Mechanism design studies how this information can be extracted, and the extent to which the information revelation problem constraints the ways in which social decisions can respond to individual preferences. Applications of mechanism design ; 1- Design of voting procedure 2- writing of contracts among parties 3- construction of procedures for deciding upon public projects or environmental standards are. MASCOLLEL,WINSTON,CREEN1Incentives and Mechanism Design CH 23

2. Incentives and Mechanism Design The mechanism design problem Assumptions; I agents i = 1,2,3 …….I. These agents must make a collective action from a set of X possible alternatives. Prior to the choice each agent i privately observes a alternatives in X. In other words, agent i privately observe a signal θ i that determines his preferences. We will refer to θ i as agent I’s type. The set of possible types for agent i is denoted by  Agent’s i utility = u i ( x, θ i ). He is a expected utility maximize. The ordinal preference relation over pairs of alternatives in X that is associated with utility function u i ( x, θ i ) is denoted by ≿ i (θ i ). Note that because θ i is observed only by agent i, in the language of game theory it is characterized by incomplete information. We assume that the agents are drawn from a commonly known prior distribution. In this way we denote a profile of agents’ type by θ = (θ 1, θ 2,,,, θ I ). MASCOLLEL,WINSTON,CREEN2Incentives and Mechanism Design CH 23

3. Incentives and Mechanism Design θ = (θ 1, θ 2,,,, θ I ) = profile of agent types. = probability density over the possible realization of θ The probability density of as well as the set and the utility function u i (., θ i ) are the common knowledge among the agents but the specific value of each agent i type is observed only by i. Because the agents’ preference depend on the realization of θ = (θ 1, θ 2,,,, θ I ), the agents may want the collective decision to depend on θ. To capture this formally we need to introduce the concept of social welfare function. Definition 1 ; a social welfare function is a function f ; → X that, for each possible profile of the agents’ types (θ 1, θ 2,,,, θ I ), assigns a collective choice f(θ 1, θ 2,,,, θ I ) є X. Definition 2 ; a social welfare function f: → X s ex-post efficient ( or Paretian) if for no profile θ = (θ 1, θ 2,,,, θ I ) is there an xєX such that u i ( x, θ i ) ≥ u i ( f(θ), θ i ) for every i, and u i ( x, θ i ) ≥ u i ( f(θ), θ i ) for some i. MASCOLLEL,WINSTON,CREEN3Incentives and Mechanism Design CH 23

4. Incentives and Mechanism Design Definition 2 says that a social welfare function is ex post efficient if it selects, for every profile θ = (θ 1, θ 2,,,, θ I ), an alternative f(θ) є X that is pareto optimal given the agents’ utility functions u 1 (.,θ 1 ),...u I (.,θ I ). The problem faced by agent is that the θ i ‘s are not publicly observable and so for the social choice f(θ 1, θ 2,,,, θ I ) to be chosen when the agents’ types are (θ 1, θ 2,,,, θ I ), each agent must be relied upon to disclose his type θ i. However agent might not find it to be in his best interest to reveal this information truthfully. Example 1 ; An abstract Social Choice Setting. In most of the cases we are given a set of X and, for each agent i, a set of possible rational preference orderings on X. Suppose that X = { x, y, z } and that I=2. Individual 1 has one possible type so that, and that agent 2 has two possible types:. The agents possible preference ordering are as follows : MASCOLLEL,WINSTON,CREEN4Incentives and Mechanism Design CH 23

5. Incentives and Mechanism Design X Z Y Y Y X Z X Z Now suppose that the agents wants to implement the ex post efficient social choice function f(.) with and. If so. Then agent 2 must be relied upon to truthfully reveal his preferences. But it is apparent that he will not find it in his interest to do so: when, agent 2 will wish to lie and claim that his type is, because in θ ’ 2 he could contribute less, since X and Y have lower rank in his preferences. It may be very difficult for a social function always to induce the agents to reveal their preferences truthfully. We will see a formal illustration of this point in Gibbard-atterthwaite theorem. MASCOLLEL,WINSTON,CREEN5Incentives and Mechanism Design CH 23

6. Incentives and Mechanism Design Example 2 : a pure exchange economy : Consider a pure exchange economy with L goods and I consumers in which agent i has consumption set R L + and endowment vector w i (w 1i,,,,,w Li ) >0. The set of alternatives are as follows : X = { x 1,…..x I ): x i є R L + and ∑x l i ≤ ∑w li for l = 1,….L. We will assume that R i which is each consumer i’s set of possible preference relations over alternatives in X is a subset of R E the set of individualistic, monotone, and convex preference relation on X. Suppose that I=2 (there are two consumers), and consumer 1 has only one possible type. So that,and R 1 = and that the consumer 2 have R 2 = R E. We try to implement a social welfare function that, for each pair of chooses a Walrasian equilibrium allocation. As the following figure illustrate consumer 2 will not generally find it optimal to reveal his preferences truthfully. MASCOLLEL,WINSTON,CREEN6Incentives and Mechanism Design CH 23

7. Incentives and Mechanism Design MASCOLLEL,WINSTON,CREEN7Incentives and Mechanism Design CH 23 o1o1 o2o2 w oc 1 oc’ 2 oc” 2 When consumer’2 offer curve is oc ’ 2 and his type is θ’ 2, he will always claim that his type is θ ” 2 and obtain a higher level of utility.

8. Incentives and Mechanism Design Example 3 : A public project : I agents must decide whether to undertake a public project which should be founded by themselves. An outcome is a vector x=(k, t 1,….t I ), where k=1 if project is built and k=0 if project is not. t i is money transfer from agent i ( t i 0). The cost of project is c ≥0 and there is no source of outside funding for the project. So the set of feasible alternative projects for the I agents is : X={(k, t 1,…t I ): kє{0,1}, t i є R for all i, and ∑t i ≤ -ck}. Note that in this relation ∑t i is income gathered through taxpayers and –ck is the cost of project. If the project is going to be built (k=1), then the following relation should hold : c + ∑t i ≤0. We assume that the Burnolli utility function has the following form ; u(x i, θ i ) = θ i k + ( +t i ) where is agent’s i initial endowment of the numeraiar (money) and θ i є . θ i is the agent’s i willingness to pay for the project. the social function f(θ) = (k(θ), t 1 (θ), t 2 (θ), …. t I (θ)) is ex post efficient if for all θ, k(θ)=1 if ∑ i (θ) ≥ c, and k(θ)=0 otherwise, and ∑ i t i (θ)= -ck(θ). MASCOLLEL,WINSTON,CREEN 8Incentives and Mechanism Design CH 23

9. Incentives and Mechanism Design Suppose that the agents want to implement a social choice function that satisfies the above relation and in which the egalitarian contribution rule is followed. In this way we should have : transfer from agent i = t i (θ) = - ( c/I)k( θ) = equal share of the cost of project, k=1). Suppose that Θ i = for i ≠ 1 so that all agents other than agent 1 have preferences that are known. Suppose that : This implies that the agent’s one type is critical for whether the bridge should be built or not. If θ 1 ≥ c - ∑ i≠1 it is and if θ 1 ≥ c - ∑ i≠1 it is not and  the sum of utilities of agents 2,.. I, should be strictly greater than their transfer payment if the bridge is built : Let us examine agent1’s incentive for truthfully revealing his type when ; MASCOLLEL,WINSTON,CREEN9Incentives and Mechanism Design CH 23

10. Incentives and Mechanism Design If the agent 1 reveal his true preferences, the bridge will be built because Agent’s 1’s utility in this case is : But for small enough, this is less than, which is agent 1’s utility if he instead claims that =0, a claim that results in the bridge not being built. Thus, agent 1 will prefer not to tell the truth. Intuitively, under this allocation rule, when agent 1 causes the bridge to be built he has a positive externality on the other agents because he fails to internalize this effect, he has an incentive to understate his benefit from the project. MASCOLLEL,WINSTON,CREEN10Incentives and Mechanism Design CH 23

11. Incentives and Mechanism Design Example 4 : Allocation of a single unit of an indivisible private good. A single unit of indivisible good to be allocated to one of I agents. Outcome can be represented by : x=(y 1,….,y I, t 1,….t I ). If y i =1, agent i gets the money and if y i =0 he does not. Then, t i is the monetary transfer from agent i. Then the feasible set of alternatives is; X= {(y 1,….,y I, t 1,….t I ) : y i є {0,1} and t i є R for all I, ∑ i y i =1, ∑ i t i ≤ 0 }. Type θ i ‘s utility function = u i (x, θ i ) = θ i y i + ( + t i ). Where ; = initial endowment of money ( numariair good). And t i є R can be viewed as agent’s I valuation of good. Set of possible valuation for agent i can be shown by R In this context the social choice function : f(θ) = ( y 1 (θ),…y I (θ), t 1 (θ),… t I (θ) is ex-post efficient if it always allocate the good to the agent who has the highest valuation and involves no waste of the money that is ; For all MASCOLLEL,WINSTON,CREEN11Incentives and Mechanism Design CH 23

12. Incentives and Mechanism Design y i (θ) (θi - max{θ 1,.. θ I })=0 for al i, since ; if i gets the commodity then y i =1, and θ i = max{θ 1,.. θ I } and if i does not get the commodity then y i = 0, and θ i ≠ max{θ 1,.. θ I }, we should also have no waste, which is ∑ i t i (θ) = 0. This means that sum of monetary transfer should be equal to zero ( tax collected = expenditure). Two special cases that received a great deal of attention in the literature ; 1- Bilateral trade ; I=2, in which agent 1 is the initial owner of the good (seller) and agent 2 is the potential purchaser ( buyer). When or minimum value for buyer > maximum value of seller there will be fruitful trade regardless of the values of θ. When or minimum value for seller > maximum value of buyer, there will not be any trade regardless of the value of θ. When and there might be fruitful trade depending upon the values of θ. 2- Auction setting : one agent, namely agent 0, is the auctioneer or seller of the good and assumed to have a known value. The others are potential buyers ( the bidders ). MASCOLLEL,WINSTON,CREEN12Incentives and Mechanism Design CH 23

13. Incentives and Mechanism Design Consider the auction setting with two potential buyers (I=2, i=1,2). In this example we assume that both buyers’ valuation of θ i which is privately observed are drawn independently from the uniform distribution on [ 0,1] and this fact is the common knowledge among the agents. Consider the social choice function : f(y 0 (θ), y 1 (θ), y 2 (θ), t 0 (θ), t 1 (θ), t 2 (θ) ), in which ; y 1 (θ) = 1, if θ 1 ≥ θ 2 and y 1 (θ) = 0, if θ 1 < θ 2 y 2 (θ) = 1, if θ 1 < θ 2 and y 2 (θ) = 0, if θ 1 ≥ θ 2 y 0 (θ) =0 for all θ t 1 (θ) = - θ 1 y 1 (θ) t 2 (θ) =- θ 2 y 2 (θ) t 0 (θ) = - (t 1 (θ) + t 2 (θ) ) In this social choice function the seller gives the good to the buyer with the highest valuation and this buyer gives the seller a payment equal to his evaluation. The other makes no transfer payment. If buyer 2 always announces his true value (θ 2 ), will buyer 1 find it optimal to do so. MASCOLLEL,WINSTON,CREEN13Incentives and Mechanism Design CH 23

14. Incentives and Mechanism Design The buyer 1 problem is to choose a value for θ such as as low as possible. Putting this maximization problem in an uncertainty context the agent one problem is to maximize : or → We see that if buyer 2 always tells the truth (announce θ 2 ), truth telling is not optimal for buyer 1, since true value for agent 1 is θ 1 and agent 1 is announcing and. The same analysis will be true for agent 2. Intuitively for this social welfare function a buyer has incentive to underestimate his valuation so that to lower the transfer he must make in the event that he has the highest announced valuation and gets the good. The cost of doing this action is to get the good less often, but this cost is worth doing to some degree in most of the time. Now suppose that we have a social welfare function the same as before but with the following transfer functions ; MASCOLLEL,WINSTON,CREEN14Incentives and Mechanism Design CH 23

15. Incentives and Mechanism Design t 1 (θ) = - θ 2 y 1 (θ) t 2 (θ) =- θ 1 y 2 (θ) t 0 (θ) = - (t 1 (θ) + t 2 (θ) ) In this type of social choice function instead of buyer 1 paying the seller an amount equal to his own evaluation θ 1 if he wins the object, he will pay t 1 (θ) = - θ 2, (y 1 (θ)=1) the same is true for buyer 2. He will pay t 2 (θ) =- θ 1 (y 2 (θ) =1). That is the agent is paying an amount to the second- highest valuation. Now consider buyer 1’s incentive for truth telling. If buyer 2 announces his valuation to be, buyer 1 can receive a utility of by truthfully announcing that his valuation is θ 1. Since he has to pay and his valuation is. For any other announcement of θ 1 the agent gain no utility (or the resulting utility is the same ) if he announces a valuation as great as and resulting utility may be zero when he announce an amount below since he will loose utility. Now if buyer’2 announcement is, buyer’s 1 utility is zero if he reveals his true valuation. Since he will loose utility. However buyer one can receive only a negative utility by making a false claim that gets him the good ( a claim that his evaluation is at least ) MASCOLLEL,WINSTON,CREEN 15Incentives and Mechanism Design CH 23

16. Incentives and Mechanism Design. We can conclude that truth telling is optimal for agent 1 regardless of what buyer 2 announces. A similar conclusion could be drawn for buyer 2. Thus this social welfare function is implementable even though the buyers ‘ valuations are private information. The examples 1 through 4 suggest that when agent’s type are privately observed the information revelation problem may constrain the set of social choice function that can be successfully implemented. So the type of social choice function is the important question To answer this question, we need in principle to begin by thinking of all the possible ways in which a social function might be implemented. In examples 1 to 4 the agent i is asked to directly reveal θ i, and the given the announcement, the alternative is chosen. But, a given social choice function might be indirectly implemented by having the agents interact through some type of institution in which there are rules governing the actions that agents may take and how these actions translate into a social outcome. Examples 5 and 6 illustrates this. MASCOLLEL,WINSTON,CREEN16Incentives and Mechanism Design CH 23

17. Incentives and Mechanism Design Example 5 ; First-Price Sealed-Bid Auction In this auction each buyer i is allowed to submit a sealed bid, b i ≥0. The bids are opened and the buyer with the highest bid gets the good and pays an amount equal to his bid to the seller. Suppose that there are two potential buyers and each θ i is independently drawn from the uniform distribution on [ 0,1]. We are looking for the equilibrium in which each buyer’s strategy b i (.) takes the form as ; b i (θ i ) = α i θ i for α i є [0,1 ]. The optimization problem for buyer 1 is ; Max b1≥0 (θ 1 - b 1 ) prob (b 2 (θ 2 ) ≤ b 1 ). Since buyer’2 highest possible bid is α 2 ( when θ 2 has the highest value equal to1 ), it is evident that buyer 1 would never bid for more than α 2 Now b 2 (θ 2 ) ≤ b 1 if and only if θ 2 ≤ (b 1 ) / α 2. Then buyer’s 1 problem is to Max b1є[0,α2 ] (θ 1 - b 1 ) (b 1 / α 2 ) since prob (b 2 (θ 2 ) ≤ b 1 ) = (b 1 / α 2 ). Solution to this problem is ; b 1 (θ 1 ) = (1/2) θ 1 if (1/2) θ 1 ≤ α 2 b 1 (θ 1 ) = α 2 if (1/2) θ 1 > α 2 since the highest value for b 1 is α 2. MASCOLLEL,WINSTON,CREEN17Incentives and Mechanism Design CH 23

18. Incentives and Mechanism Design We could have the same solution for the buyer 2 b 2 (θ 2 ) = (1/2) θ 2 if (1/2) θ 2 ≤ α 1 b 2 (θ 2 ) = α 2 if (1/2) θ 2 > α 1 Letting α 1 = α 2 = 1/2 we can see that the strategies b i (θ i ) = (1/2) θ i constitute a Baysian Nash equilibrium for this auction. Thus there is a Baysian Nash equilibrium of this first-price sealed –bid auction that indirectly yields the outcomes specified by the social choice function f(y 0 (θ), y 1 (θ), y 2 (θ), t 0 (θ), t 1 (θ), t 2 (θ) ), in which ; y 1 (θ) = 1, if θ 1 ≥ θ 2 and y 1 (θ) = 0, if θ 1 < θ 2 y 2 (θ) = 1, if θ 1 < θ 2 and y 2 (θ) = 0, if θ 1 ≥ θ 2 y 0 (θ) =0 for all θ t 1 (θ) = - (1/2) θ 1 y 1 (θ) t 2 (θ) =- (1/2) θ 2 y 2 (θ) t 0 (θ) = - (t 1 (θ) + t 2 (θ) ) MASCOLLEL,WINSTON,CREEN18Incentives and Mechanism Design CH 23

19. Incentives and Mechanism Design Example 6 : Second-price Sealed-bid Auction. Each potential buyer i is allowed to submit a sealed bid, b i ≥ 0. The bids are then opened and the buyer with the highest bid gets the good, but now he pays the seller an amount equal to the second-highest bid. The strategy b i (θ i ) = θ i, for all θ i є [ 0,1] is a weakly dominant strategy for each buyer since the winner will get b i (θ i ) = θ i but will pay less than θ i (θ j < θ i )and it implements the social choice function ; f(y 0 (θ), y 1 (θ), y 2 (θ), t 0 (θ), t 1 (θ), t 2 (θ) ), in which ; y 1 (θ) = 1, if θ 1 ≥ θ 2 and y 1 (θ) = 0, if θ 1 < θ 2 y 2 (θ) = 1, if θ 1 < θ 2 and y 2 (θ) = 0, if θ 1 ≥ θ 2 y 0 (θ) =0 for all θ t 1 (θ) = - θ 2 y 1 (θ) t 2 (θ) =- θ 1 y 2 (θ) t 0 (θ) = - (t 1 (θ) + t 2 (θ) ) MASCOLLEL,WINSTON,CREEN19Incentives and Mechanism Design CH 23

20. Incentives and Mechanism Design Examples 5 and 6 illustrates that, as a general matter, we need to consider not only the possibility of directly implementing social choice function by asking the agents to reveal their types but also their indirect implementation through thr design of institutions in which the agent s interact. The formal representation of such an institution is known as mechanism. Definition 3 : A mechanism Γ = ( S 1,…, S I, g(.)) ia a collection of I strategy sets ( S 1,.. S I ) and an outcome function g: S 1  ….  S I → X A mechanism can be viewed as an institution with rules governing the procedure for making the collective choice. The allowed action of each agent i are summarized by the strategy set S i, and the rule for how agents’ actions get turned into a social choice is given by the outcome function g(.). The mechanism Γ combined with possible types ( Θ 1,,,,Θ I ), probability density  (.), and a Bunolli utility functions ( u 1 (.),…u I (.) ) defines a Bayesian game of incomplete information. MASCOLLEL,WINSTON,CREEN20Incentives and Mechanism Design CH 23

21. Incentives and Mechanism Design For the auction setting, the first-price sealed- bid Auction is the mechanism in which S i =  + for all i and, given the bids ( b 1, …b I ) є  I +, the outcome function g( b 1,… b I ) = ( {y i (b 1,… b I )} I i=1, {t i (b 1,… b I )} I i=1 ) is such that ; y i (b 1,… b I ) =1 if and only if I = Min {j: b j = Max {b 1,… b I }}, t i (b 1,… b I ) = -b i y i (b 1,… b I ). In the second-price sealed-bid auction, we have the same strategy sets and functions y i (.), but instead t i (b 1,… b I ) = - Max {b j j≠i }y i (b 1,… b I ). Definition 4 ; the mechanism Γ = (S 1,… S I, g(.)) implements social choice function f(.) if there is an equilibrium strategy profile ( s * 1 (.),… s * I (.)) of the game induced by Γ such that g( s * 1 (θ 1 ),… s * I (θ I ))=f(θ 1, …. θ I ) for all ( θ 1,… θ I ) є Θ 1  …  Θ I. What we mean by equilibrium will be explained under two topics ; first ; dominant strategy equilibrium and second : Baysian Nash equilibrium. Note that mechanism Γ may have more than one equilibrium, but definition 4 requires that only one of them induce outcomes in accord with f(.). Definition 4 assumes that the agent will play that equilibrium that the mechanism wants. MASCOLLEL,WINSTON,CREEN21Incentives and Mechanism Design CH 23

22. Incentives and Mechanism Design The implementation of all social choice function that are implementable may seem a difficult task. The revelation principle tells us that we can often restrict attention to the very simple type of mechanisms in which each agent is asked to reveal his type. These are known as the direct revelation principle. Definition 5 : a direct revelation principle is a mechanism in which S i = Θ i for all I and g(θ)=f(θ) for all θ є Θ 1,….Θ I. Moreover as we shall see, the revelation principle also tells us that we can further restrict our attention to direct revelation mechanism in which truth telling is an optimal strategy for each agent. This fact motivates the notion of truthful implementation ; Definition 6: the social choice function f(.) is truthfully implementable (or incentive compatible) if the direct revelation mechanism Γ= (Θ 1,….Θ I ), f(.))has an equilibrium ( s * 1 (.),… s * I (.)) in which s * i (θ i )=θ i for all θ i є Θ i and all i=1,….I : that is, if truth telling by each agent i constitutes an equilibrium of Γ=(Θ 1,….Θ I, f(.)) To offer a hint as to we may be able to restrict attention to direct revelation mechanisms that induce truth telling, we briefly verify that the social choice MASCOLLEL,WINSTON,CREEN22Incentives and Mechanism Design CH 23

23. Incentives and Mechanism Design functions that are implemented indirectly through the first-price and second-price sealed bid auctions in examples 5 and 6 can e truthfully implemented using a direct revelation principle. Example 7 : Truthful Implementation of the Social Choice Function Implemented by the First-Price Sealed-Bid Auction. When facing the direct revelation mechanism (Θ 1,….Θ I, f(.)) with f(y 0 (θ), y 1 (θ), y 2 (θ), t 0 (θ), t 1 (θ), t 2 (θ) ), in which ; y 1 (θ) = 1, if θ 1 ≥ θ 2 and y 1 (θ) = 0, if θ 1 < θ 2 y 2 (θ) = 1, if θ 1 < θ 2 and y 2 (θ) = 0, if θ 1 ≥ θ 2 y 0 (θ) =0 for all θ t 1 (θ) = - (1/2) θ 1 y 1 (θ) t 2 (θ) =- (1/2) θ 2 y 2 (θ) t 0 (θ) = - (t 1 (θ) + t 2 (θ) ) Buyer 1’s optimal announcement θ * 1 when he has the type θ 1 should solve Max (θ 1 – ½ θ * 1 )prob(θ 2 ≤ θ * 1 ) with respect to θ * 1 or Max (θ 1 – ½ θ * 1 )θ * 1 with respect to θ * 1 → θ * 1 = θ 1 → true value MASCOLLEL,WINSTON,CREEN23Incentives and Mechanism Design CH 23

24. Incentives and Mechanism Design The first order condition for this problem gives θ * 1 = θ 1. So truth telling is buyer’s 1 optimal strategy given that buyer 2 always tells the truth. A similar conclusion follows for buyer 2. Thus, the social choice function implemented by the first-price sealed-bid auction can also be truthfully implemented through a direct revelation mechanism. That is the social choice function chosen is incentive compatible. C - Dominant Strategy Equilibrium ; θ =(θ 1, …. θ I ) = vector of agent type drawn from the set Θ=(Θ 1  …  Θ I ) according to a probability density function  (.). U i = u i (x, θ i ) = agent’s I Bernoulli utility function over all alternatives of X given his type θ i. θ -i =(θ 1, …. θ -i, θ i+1,….,θ I ), θ = (θ i, θ -i ), Θ -i = (Θ 1  …  Θ i-1  Θ i+1  …  Θ I ). A the mechanism Γ = (S 1,… S I, g(.)) is a collection o I sets S 1,… S I. Each S i containing agent i’s possible actions (s i ) ( or plans of actions), and an outcome function g: S→X, where S= S 1 ,…  S I. MASCOLLEL,WINSTON,CREEN24Incentives and Mechanism Design CH 23

25. Incentives and Mechanism Design s i-1 = ( s 1,…. s i-1, s i+1,..., s I ), s = (s i, s -i ), S -1 = S 1  ….  S i-1  S i+1  ….  S I. A strategy is a weakly dominant strategy for a player in a game if it gives him at least as large as payoff as any of his other possible strategies for every possible strategy that his rivals might play. In the present incomplete information environment, strategy s i : Θ i → S i is a weekly dominant strategy for a agent i in mechanism Γ = (S 1,… S I, g(.)) if, for all θ i є Θ i and all possible strategies for agents j≠ i, s i-1 (.) = (s 1 (.)…. s i-1 (.),s i+1 (.),..., s I (.)), we have : the above condition holding for all s -i (.) and θ i is equivalent to the condition that, for all θ i є Θ i we should have This leads to the definition C.1 : MASCOLLEL,WINSTON,CREEN25Incentives and Mechanism Design CH 23

26. Incentives and Mechanism Design Definition C.1 : the strategy ( s * 1 (.),… s * I (.)) is a dominant strategy equilibrium of mechanism Γ = (S 1,… S I, g(.)) if, for all i and θ i ∈ Θ i : for all S ’ i ∈ S i, S -i ∈ S -i Definition C.2 : the mechanism Γ = (S 1,… S I, g(.)) implements the social choice function f(.) in domiant strategies if there exist a dominant strategy equilibrium of Γ, s * (.) = ( s * 1 (.),… s * I (.)), such that g(s * (θ)) = f(θ) for all θ ∈ Θ. If we can find a mechanism Γ = (S 1,… S I, g(.)) that implements f(.) in dominant strategies, then this mechanism implements f(.) in a very strong and robust way. Since, 1- a rational agent who has a dominant strategy will play it. 2- unlike the equilibrium strategies in Nash-related equilibrium concepts, a player need not correctly forecast his opponents’ play to justify his play of a dominant strategy. 3- although we have assumed that the agents know the probability density  (.) over realization of the types (θ 1, …. θ -i, θ i+1,….,θ I ), and hence can deduce MASCOLLEL,WINSTON,CREEN26Incentives and Mechanism Design CH 23

27. Incentives and Mechanism Design The correct conditional probability distribution over realizations of θ i, if Γ implies f(.) in dominant strategies, this implementation will be robust even if agents have incorrect, and perhaps even contradictory, beliefs about this distribution. In particular, agent’s i beliefs regarding the distribution of θ i do not affect the dominance strategy of s * i (.). 4- it follows that if Γ implements f(.) in dominant strategies then it does so regardless of the probability density  (.). Thus the same mechanism can be used to implement f(.) for any  (.).one advantage of this is that the government need not know  (.) to successfully implement f(.). For the dominant strategy implementation, we do not need to consider all possible mechanisms, but it suffice to ask whether a particular f(.) is truthfully implementable as discussed in definition C.3 Definition C.3 : the social function f(.) is truthfully implementable in dominant strategies ( or dominant strategy incentive compatible, or strategy- proof, or straightforward ) if s * i (θ i ) = θ i for all θ i ∈ Θ i, and i = 1,…….I is a dominant strategy equilibrium of the direct revelation mechanism : Γ= (Θ 1,….Θ I ), f(.)). That is for all i and θ i ∈ Θ i ; u i (f(θ i, θ -i ), θ i ) ≥ u i (f(θ 0 i, θ -i ), θ i ) for all θ 0 i ∈ Θ i and all θ -i ∈ Θ -i. The equilibrium dominant strategy in social choice function should have higher utility for the agent than any other strategy in the social function MASCOLLEL,WINSTON,CREEN27Incentives and Mechanism Design CH 23

28. Incentives and Mechanism Design the ability to restrict our inquiry, without loss of generality, to the question of whether f(.) is truthfully implementable is a consequence of what is known as revelation principle for dominant strategies. Preposition C.1 : (The Revelation Principle of Dominant Strategies ), suppose that there exist a mechanism Γ = (S 1,… S I, g(.)) that implements the social choice function f(.) in dominant strategies. Then f(.) is truthfully implementable in dominant strategies. suppose that the indirect mechanism Γ = (S 1,… S I, g(.)) implements f(.) in dominant strategies, and that in this indirect mechanism each agent i finds playing s * i (θ i ) when his type is θ i better than playing any other s i ∈ S i for any choice s -i ∈ S -i by agent j≠i. Now consider altering this simple mechanism by introducing a mediator who says to each agent i : you tell me your type, and when you say your type is θ i, will pay s * i (θ i ) for you. Clearly if s i * (θ i ) is agent i’s optimal choice for each θ i ∈ Θ i in the initial mechanism Γ for any strategies chosen by the other agents, then agent i will find telling truth to be a dominant strategy in this new scheme. But this means that we have found a way to truthfully implement f(.). MASCOLLEL,WINSTON,CREEN28Incentives and Mechanism Design CH 23

29. Incentives and Mechanism Design This implication of the revelation principle is that to identify the set of social choice functions that are implementable in dominant strategies, we need only identify those that are truthfully implementable. In other words we should check u i (f(θ i, θ -i ), θ i ) ≥ u i (f(θ 0 i, θ -i ), θ i ) for all θ 0 i ∈ Θ i and all θ -i ∈ Θ -i. This inequality is necessary and sufficient for a social choice function f(.) to be truthfully implementable in dominant strategies can be explained in terms of weak preference reversal property. u i (f(θ ’ i, θ -i ), θ ’ i ) ≥ u i (f(θ ” i, θ -i ), θ ’ i ) and u i (f(θ ” i, θ -i ), θ ” i ) ≥ u i (f(θ ’ i, θ -i ), θ ” i ) This means that agent i’s preference ranking of f(θ ’ i, θ -i ) and f(θ ” i, θ -i ) must weakly reverse when his type changes from θ ’ i to θ ” i. If this weak preference reversal property holds for all θ ’ i and θ ” i ∈ Θ i, truth telling is a dominant strategy for agent i. This weak reversal property can be successfully stated using agent’s i lower Contour set. We could define z as the lower contour set of alternative X when agent i has type θ i By; L i (x, θ i ) = {z ∈ X : u i (x, θ i ) ≥ u i ( z, θ i )}. MASCOLLEL,WINSTON,CREEN 29Incentives and Mechanism Design CH 23

30. Incentives and Mechanism Design Now we can characterize the set of social choice functions that can be truthfully implemented dominant strategies as following ; Proposition C.2 : the social choice function f(.) is truthfully implementable in dominant strategies if and only if for all i, all θ -i ∈ Θ -i, and pairs of types for agent i, {θ ’ i and θ ” i ∈ Θ -i }, we have, f(θ ” i, θ -i ) ∈ L i ( f(θ ’ i, θ -i ), θ ’ i ), and f(θ ’ i, θ -i ) ∈ L i ( f(θ ” i, θ -i ), θ ” i ) The idea behind this preference reversal characterization of the social choice function f(.) that can be truthfully implemented in dominant strategies is illustrated in the followings ; If truth telling is a weekly dominant strategy for agent 1, then when his type changes from θ ’ 1 to θ ” 1, he must experience a week preference reversal between outcomes f( θ ’ 1, θ ’ 2 ) and f( θ ” 1, θ ’ 2 ) for each possible value of θ 2 When his type is θ ’ 1 he should have more utility with f( θ ’ 1, θ ’ 2 ) outcome than with f( θ ” 1, θ ’ 2 ) outcome. And when his type is θ ” 1 he should have more utility with f( θ ” 1, θ ’ 2 ) than with f( θ ’ 1, θ ’ 2 ) outcome. MASCOLLEL,WINSTON,CREEN30Incentives and Mechanism Design CH 23 θ ”’ 2 θ ” 2 θ ’ 2 f( θ ’ 1, θ ”’ 2 ) f( θ ’ 1, θ ” 2 ) f( θ ’ 1, θ ’ 2 ) θ ’ 1 f( θ ’” 1, θ ‘“ 2 ) f( θ ” 1, θ ” 2 ) f( θ ” 1, θ ’ 2 ) θ ” 1

31. Incentives and Mechanism Design MASCOLLEL,WINSTON,CREEN31Incentives and Mechanism Design CH 23 x 2i x 1i ≿ i ( θ ’ i ) ≿ i ( θ ” i ) agent i’s preferences satisfy the single crossing property According to proposition C.2 f i (θ ” i, θ 2 ) must lie in the shaded area if truth telling is to be a dominant strategy for agent i. f i (θ ” i, θ -i ) with ≿ i ( θ ” i ) f i (θ ’ i, θ -i ) with ≿ i ( θ ” i ) f i (θ ’ i, θ -i ) with ≿ i ( θ ’ i ) f i (θ ” i, θ -i ) with ≿ i ( θ ’ i )

32. Incentives and Mechanism Design The Gibbard-Satterhwaite Theorem ; This theorem is a impossibility result similar in spirit to Arrow’s impossibility theorem. It shows that for every general class of problems there is no hope of implementing satisfactory social functions in dominant strategies. let P denote the set of all rational preference relations ≿ on X having the property that no two alternatives are indifferent, and R i = { ≿ i : ≿ i = ≿ i (θ i ) for some θ ∈ Θ i } is agent I’s set of possible ordinal preference relation over x. f(Θ) is the image of f(.) that is ;f(Θ) = { x ∈ X : f(θ) = x for some θ ∈ Θ i }. Definition C.4. The social choice function f(.) is dictatorial if there is an agent i such that, for all θ = (θ 1 …. θ I ) ∈ Θ, f( θ) ∈ { x∈ X: u i (x, θ i ) ≥ u i (y, θ i ) for all y ∈ x } in other words a social welfare function is dictatorial if there is an agent i such that f(.) always chooses one of i’s top-ranked alternatives. MASCOLLEL,WINSTON,CREEN32Incentives and Mechanism Design CH 23

33. Incentives and Mechanism Design Definition C.5 : the social choice function f(.) is monotonic if, for any θ, if θ ’ is such that L i (f(θ), θ i )  L i (f(θ), θ ’ i ) for all i [L i (f(θ), θ i ) is weekly included in L i (f(θ), θ ’ i ) for all i ], then f(θ ’ ) = f(θ) MASCOLLEL,WINSTON,CREEN33Incentives and Mechanism Design CH 23 x 1i x 2i L i (f(θ), θ –i, θ ’ i ) If f(.) is monotonic, then f(θ i, θ -i ) = f(θ ’ i, θ -i ) L i (f(θ), θ –i, θ i ) ≿ i ( θ ’ i ) ≿ i ( θ i )

34. Incentives and Mechanism Design Proposition C.3 : ( The Gibbard-Satterthwaite Theorem) Suppose that X is finite and contain at least three elements, that R i = P for all i, and that f(Θ) = X. Then the social choice function f(.) is truthfully implementable in dominant strategies if and only if it is dictatorial. It should be noted that the conclusion of proposition C.3 does not follow if X contains two elements. For example, in this case, a majority voting social choice function is both non dictatorial ( the ranking is not the will of only one person) and truthfully implementable (every one announcing the true desire) in dominant strategies. Given this negative conclusion, if we are to have any hope of implementing desirable social functions, we must either weaken the demands of our implementation concept by accepting implementation by means of less robust equilibrium notions or we must focus on more restricted environment. First we follow the case of restricted environment when preferences take a quasi- linear form, then we will study the less robust equilibrium by considering the notion of Baysian Nash equilibrium MASCOLLEL,WINSTON,CREEN34Incentives and Mechanism Design CH 23

35. Incentives and Mechanism Design Quasi-linear Environments : Groves-Clark Mechanism ; Vector x = ( k, t 1, …t I ), where k is an element of a finite set K called the “project choice” and t i ∈ R is a transfer of numerator commodity ( money) to agent i. Agent i’s qauisilinear u tility function is as follows : u i (x, θ i ) = v i (k, θ i ) + ( m 0 i + t i ) m 0 i = agent i’s endowment, there is no source of outside financing. Example C.1: A public project. Let K contains the possible set of public project ( K=0, K=1). Let c(k) the cost of project level k ∈ K. v 0 i (k, θ i ) = agent i’s gross benefit from project level k. c(k)/I = equal share for each agent’s contribution for building the project. v i (k, θ i ) = v 0 i (k, θ i ) - c(k)/I = agent’s net benefit form project level k. The t i ‘s are transfers over and above the payments c(k)/I. Allocation of a single unit of invisible private good : An invisible unit of a private good is to be allocated to one of I agents. MASCOLLEL,WINSTON,CREEN35Incentives and Mechanism Design CH 23

36. Incentives and Mechanism Design The project choice k = ( y 1 ….y I ) represents the allocation of the private good and K= { ( y 1 …. y I ) : y i ∈ { 0,1} for all i and Σ i y i =1. v i ( k, θ i ) = θ i y i agent’s i valuation function. f(.) is the social choice function in quasi-linear environment ; f(.) = ( k(.), t 1 (.),…., t I (.) ), all θ ∈ Θ, k(θ ) ∈ K Σ i t i (θ) ≤ 0 If f(.) is ex post efficient, for all θ ∈ Θ and k(θ) we should have : Σ I i=1 v i (k(θ),θ i ) ≥ Σ I i=1 v i (k,θ i ) for all k ∈ K. That is ; the valuation of agents in the outcome of project in the social choice function should be greater than the valuation of agents in any other situation of the project. A direct revelation mechanism Γ= (Θ 1,….Θ I ), f(.)) in which f(.) = ( k(.), t 1 (.),…., t I (.) ) satisfying the above inequality is known As a Groves mechanism. In this kind of mechanism, given the announcement θ -i of agents j≠ i, agent’s i transfer depends on his announced type only trough his announcement’s effect on the project choice k * (θ). Moreover the change in agent’s i transfer that results when his announcement changes the project decision k is exactly equal to he effect of this change in k on the agents j≠i. In other words the change in agent’s i transfer reflects the externality that he is imposing on the other agent. MASCOLLEL,WINSTON,CREEN36Incentives and Mechanism Design CH 23

37. Incentives and Mechanism Design Clark (Pivotal) Mechanism If for all θ -i ∈ Θ -i, the parameter k * -i ( θ -i ) satisfies : Σ i≠j v j (k * -i (θ -i ),θ j ) ≥ Σ i≠j v j (k,θ j ) as before this means that the valuation of all the agents except agent i with k * -i (θ -i ), should be greater than any other valuation of the project, if k * -i (θ -i ) is the project level that would be ex post efficient. Agent’s i transfer then will be equal to ; t i (θ) = [Σ i≠j v j (k * (θ ),θ j )] - [Σ i≠j v j (k * -i (θ -i ),θ j )] If k * (θ ) = k * -i (θ -i ), then agent’s i transfer is zero and his announcement does not change the project decision relative to what would be ex post efficient for other agents. And it is negative if it does change the project decision k * (θ ) ≠ k * -i (θ -i ), that is if the agent i is pivotal to the efficient project choice. Thus in the Clark mechanism agent i pays a tax equal to his effect on other agents if he is pivotal to the project decision, and pays nothing otherwise. It is interesting to note that in the case of allocation of a single individual unit of a private good, the Clark mechanism is precisely the social choice function implemented by the second-price sealed-bid auction. MASCOLLEL,WINSTON,CREEN37Incentives and Mechanism Design CH 23

38. Incentives and Mechanism Design If k * (θ ) is the allocation rule that gives the item to the agent with the highest valuation, then ; agent i is pivotal precisely when he is buyer with the highest valuation and his tax is equal to the second-highest valuation. It should be mentioned that the social choice functions which satisfies the condition Σ I i=1 v i (k(θ),θ i ) ≥ Σ I i=1 v i (k,θ i ) is the one which can be truthfully implementable in the dominant strategy equilibrium. It can also be shown that when all possible functions v i (.) can arise for some θ -i ∈ Θ -i, the only social choice function satisfying Σ I i=1 v i (k(θ),θ i ) ≥ Σ I i=1 v i (k,θ i ) that are truthfully implementable in dominant strategies are those in the Groves class. Groves mechanism and budget constraint Ex-post efficiency also requires that none of the numeraire be wasted, that is, the budget constraint should be satisfied ; Σ i t i (θ) =0 for all θ ∈ Θ. Unfortunately in many cases it is impossible to truthfully implement fully ex post efficient social choice functions in dominant strategies. MASCOLLEL,WINSTON,CREEN38Incentives and Mechanism Design CH 23

39. Incentives and Mechanism Design For example, it could be shown that if the set of possible types for each agent is sufficiently rich, then no social choice functions that are truthfully implementable in dominant strategies are ex post efficient. Thus it could be concluded that ; the presence of private information means that the I agents must either accept some waste of numeraire (Σ i t i (θ) < 0 ) or give up on always having an efficient project setting k(θ) that dos not satisfy Σ I i=1 v i (k(θ),θ i ) ≥ Σ I i=1 v i (k,θ i ) One special case ; There is one agent whose preference is known ( agent 0 ), and I other agents ( i=1,,,,I ) whose preferences are not known ( they are private information ). For example the simplest case is the one in which the agent 0 has no preferences over the project choice. One example could be the one in auction setting when agent 0 is the seller. When there is such an agent, ex post efficiency of the social choice function still requires that this relation be satisfied ; Σ I i=1 v i (k(θ),θ i ) ≥ Σ I i=1 v i (k,θ i ). But now ex post efficiency is compatible with any transfer function t 1 (.),…..t I (.) for the I agent with private information as long as we set t 0 (θ) = -Σ i≠0 t i (θ). That is in in this ( I + 1 ) agent setting, the Groves mechanism are expost efficient as long as we set the transfer of agent 0 to be t 0 (θ) = -Σ i≠0 t i (θ). For all θ. MASCOLLEL,WINSTON,CREEN39Incentives and Mechanism Design CH 23

40. Incentives and Mechanism Design MASCOLLEL,WINSTON,CREEN40Incentives and Mechanism Design CH 23

41. Incentives and Mechanism Design MASCOLLEL,WINSTON,CREEN41Incentives and Mechanism Design CH 23

42. Incentives and Mechanism Design MASCOLLEL,WINSTON,CREEN42Incentives and Mechanism Design CH 23