Theory of Mechanism Design Debasis Mishra1 April 14, 2020 1Economics and Planning Unit, Indian Statistical Institute, 7 Shahid Jit Singh Marg, New Delhi 110016, India, E-mail: [email protected] 2 Contents 1 Introduction to Mechanism Design7 1.0.1 Private Information and Utility Transfers................8 1.0.2 Examples in Practice........................... 10 1.1 A General Model of Preferences......................... 11 1.2 Social Choice Functions and Mechanisms.................... 14 1.3 Dominant Strategy Incentive Compatibility................... 16 1.4 Bayesian Incentive Compatibility........................ 18 1.4.1 Failure of revelation principle...................... 21 2 Mechanism Design with Transfers and Quasilinearity 23 2.1 A General Model................................. 24 2.1.1 Allocation Rules............................. 24 2.1.2 Payment Functions............................ 26 2.1.3 Incentive Compatibility.......................... 27 2.1.4 An Example................................ 27 2.1.5 Two Properties of Payments....................... 28 2.1.6 Efficient Allocation Rule is Implementable............... 29 2.2 The Vickrey-Clarke-Groves Mechanism..................... 32 2.2.1 Illustration of the VCG (Pivotal) Mechanism.............. 33 2.2.2 The VCG Mechanism in the Combinatorial Auctions......... 35 2.2.3 The Sponsored Search Auctions..................... 37 2.3 Affine Maximizer Allocation Rules are Implementable............. 38 2.3.1 Public Good Provision.......................... 40 2.3.2 Restricted and Unrestricted Type Spaces................ 41 3 3 Mechanism Design for Selling a Single Object 45 3.1 The Single Object Auction Model........................ 45 3.1.1 The Vickrey Auction........................... 45 3.1.2 Facts from Convex Analysis....................... 46 3.1.3 Monotonicity and Revenue Equivalence................. 49 3.1.4 The Efficient Allocation Rule and the Vickrey Auction........ 53 3.1.5 Deterministic Allocations Rules..................... 54 3.1.6 Individual Rationality.......................... 55 3.1.7 Beyond Vickrey auction: examples................... 56 3.1.8 Bayesian incentive compatibility..................... 58 3.1.9 Independence and characterization of BIC............... 59 3.2 The One Agent Problem............................. 62 3.2.1 Monopolist problem........................... 65 3.3 Optimal Auction Design............................. 70 3.4 Correlation and full surplus extraction..................... 79 4 Redistribution mechanisms 85 4.1 A model of redistributing a single object.................... 86 4.2 Characterizations of IC and IR constraints................... 87 4.3 Dissolving a partnership............................. 89 4.3.1 Corollaries of Theorem 13........................ 93 4.4 Dominant strategy redistribution........................ 96 4.5 The dAGV mechanism.............................. 101 5 Multidimensional Mechanism Design 105 5.1 Incentive Compatible Mechanisms........................ 106 5.1.1 An illustration.............................. 106 5.2 The Implementation Problem.......................... 113 5.3 Revenue Equivalence............................... 119 5.4 Optimal Multi-Object Auction.......................... 122 6 Extensions 129 6.1 Classical Preferences............................... 129 4 6.1.1 Type Spaces with Income Effects.................... 130 6.1.2 Mechanisms and Incentive Compatibility................ 133 6.1.3 Vickrey Auction with Income Effect................... 135 6.2 Interdependent valuations............................ 139 6.2.1 Mechanisms and Ex-post Incentive Compatibility........... 140 6.2.2 Efficiency: Impossibility and Possibility................. 141 7 The Strategic Voting Model 147 7.1 The Unrestricted Domain Problem....................... 147 7.1.1 Examples of Social Choice Functions.................. 149 7.1.2 Implications of Properties........................ 150 7.1.3 The Gibbard-Satterthwaite Theorem.................. 153 7.1.4 Proof of the Gibbard-Satterthwaite Theorem.............. 154 7.2 Single Peaked Domain of Preferences...................... 160 7.2.1 Possibility Examples in Single-Peaked Domains............ 163 7.2.2 Median voter social choice function................... 164 7.2.3 Properties of Social Choice Functions.................. 165 7.2.4 Characterization Result......................... 168 7.3 Randomized Social Choice Function....................... 173 7.3.1 Defining Strategy-proof RSCF...................... 173 7.3.2 Randomization over DSCFs....................... 176 7.3.3 The Counterpart of Gibbard-Satterthwaite Theorem.......... 177 8 Matching Theory 181 8.1 Object Assignment Model............................ 182 8.1.1 The fixed priority mechanism...................... 182 8.1.2 Top Trading Cycle Mechanism with Fixed Endowments........ 189 8.1.3 Stable House Allocation with Existing Tenants............. 193 8.1.4 Generalized TTC Mechanisms...................... 197 8.2 The Two-sided Matching Model......................... 199 8.2.1 Stable Matchings in Marriage Market.................. 200 8.2.2 Deferred Acceptance Algorithm..................... 201 5 8.2.3 Stability and Optimality of Deferred Acceptance Algorithm...... 203 8.2.4 Unequal number of men and women.................. 208 8.2.5 Strategic Issues in Deferred Acceptance Algorithm........... 209 8.2.6 College Admission Problem....................... 213 6 Chapter 1 Introduction to Mechanism Design Consider a seller who owns an indivisible object, say a house, and wants to sell it to a set of buyers. Each buyer has a value for the object, which is the utility of the house to the buyer. The seller wants to design a selling procedure, an auction for example, such that he gets the maximum possible price (revenue) by selling the house. If the seller knew the values of the buyers, then he would simply offer the house to the buyer with the highest value and give him a \take-it-or-leave-it" offer at a price equal to that value. Clearly, the (highest value) buyer has no incentive to reject such an offer. Now, consider a situation where the seller does not know the values of the buyers. What selling procedure will give the seller the maximum possible revenue? A clear answer is impossible if the seller knows nothing about the values of the buyer. However, the seller may have some information about the values of the buyers. For example, the possible range of values, the probability of having these values etc. Given these information, is it possible to design a selling procedure that guarantees maximum (expected) revenue to the seller? In this example, the seller had a particular objective in mind - maximizing revenue. Given his objective he wanted to design a selling procedure such that when buyers participate in the selling procedure and try to maximize their own payoffs within the rules of the selling procedure, the seller will maximize his expected revenue over all such selling procedures. The study of mechanism design looks at such issues. A planner (mechanism designer) needs to design a mechanism (a selling procedure in the above example) where strategic agents can interact. The interactions of agents result in some outcome. While there are several possible ways to design the rules of the mechanism, the planner has a particular 7 objective in mind. For example, the objective can be utilitarian (maximization of the total utility of agents) or maximization of his own utility (as was the case in the last example) or some fairness objective. Depending on the objective, the mechanism needs to be designed in a manner such that when strategic agents interact, the resulting outcome gives the desired objective. One can think of mechanism design as the reverse engineering of game theory. In game theory terminology, a mechanism induces a game-form whose equilibrium outcome is the objective that the mechanism designer has set. 1.0.1 Private Information and Utility Transfers The primitives a mechanism design problem are the set of possible outcomes or alternatives and preferences of agents over the set of alternatives. These preferences are not known to the mechanism designer. Mechanism design problems can be classified based on the amount of information asymmetry present between the agents and the mechanism designer. 1. Complete Information: Consider a setting where an accident takes place on the road. Three parties (agents) are involved in the accident. Everyone knows perfectly who is at fault, i.e., who is responsible to what extent for the accident. The traffic police comes to the site but does not know the true information about the accident. The mechanism design problem is to design a set of rules where the traffic police's objective (to punish the true offenders) can be realized. The example given here falls in a broad class of problems where agents perfectly know all the information between themselves, but the mechanism designer does not know this information. This class of problems is usually termed as the implementation problem. It is usually treated separately from mechanism design because of strong requirements in equilib- rium properties in this literature. We will not touch on the implementation problem in this course. 2. Private Information and Interdependence: Consider the sale of a single object. The utility of an agent for the object is his private information. This utility information may be known to him completely, but usually not known to other agents and the mechanism designer. There are instances where the utility information of an agent