CS294 P29: Algorithmic Game Theory November 7, 2011Lecture 9: Mechanism DesignLecturer: Christos Papadimitri ou Scribe: Yusef Shafi, Michael Landis9.1 Mechanism DesignIn Lecture 8, we saw how Social Choice Theory (SCT) leads to Mechanism Design (MD): In SCT,we want to recon ci l e the various preferences of the agents. In MD, we don’t know these preferences,but we have an objective that we wish to maximize (typically but not e x cl us i vely, total welfare), hadwe known them. We construct a game whereby the outcome wil l be chosen. This game always hasa dominating outcome, and this outcome is the optimum of our objective for the players’ utilities.Vickrey’s auction is an interesting example: It compels the agents to reveal their true valuationsand achieves the social optimum (the agent who wants the i t em most gets it). Here we look at thegeneral setting.Consider n players and a set of A alternative outcomes. Each player has a private utility function,ui∈ U : A �→ R,whereui(a) denotes the worth of outcome a ∈ A to player i. U is some set ofutility functions. There is also the designer’s objective C : A × Un�→ R, assigning a real numberto all possible situations (utilities for the players and outcomes). One important example is socialwelfare: f(a, u1,...,un)=�iui(a).We are interested in designin g a game, G. Each player’ s strat e gy se t is U — that is, everyplayer chooses a utility function (for example, her true utility, or a fictional one). We also designa mapping f : Un�→ A, mapping all poss i bl e plays to out com es in A.What are the payoffs of the game?uGi(v1,...,vn)=ui(f(v1,...,vn)),that is, if everybody plays vi(some utility in U), then the ith player gets a payoff equal to h er owntrue utility for the out come that results, th r ough the designed map f, from these choices.Since G depends on the utilities, the designer does not know G. The designer wants G to havea dominant strategy play (v1,...,,vn) such that f (v1,...,vn) ∈ arg maxa∈AC(a, u1,...,un). TheRevelation Principle (see below) says, we might as well assume that the play is (u1,...,un), that is,everybody “tells the truth.”A dominant strategy means∀i∀ui∀u�i∀u−i: uGi(ui,u−i) ≥ uGi(u�i,u−i)However, we conclude th e following from what we learned in Lecture 8.Theorem 9.1. Arrow’s Impossibility Theorem ⇒ Gilbert-Satterthwai te ⇒ our ideal game designmechanism is impossible for |A|≥3.9.2 Restricted Domains and the Vickery’s AuctionGilbert-Satterthwaite holds when U : A �→ R, the most general form of uti li ty function s. But whathappens if we create restrictions on these function s? What opportunities arise? For semilinear9-1Lecture 9: Mechanism Design 9-2domains A =(A0, Rn), which are the basic outcomes and the payments by agents, respectively.ui(a, p1,...,pn)=vi(a) − pi,wherevi(a) is the private value individual i holds for outcome a, andpiis the payment individual pays. The game mechanism, f(v1,...,vn)=(a, p1,...,pn), is designedto maximize some objective C(a, u1,...,un) as defin ed by the des i gne r.For example, take Vickery’s auction, which attempts to maximize social welfare. Recall G = Si(by the direct revelation mechanism), where Siis equal to all possible vi: A �→ R. So, for Vickrey’sauction, we define a mechanism, f : Vn�→ (A, p1,...,pn), where V = A �→ R.For a single-item auction, A =1, 2,...,n andvi(j)=�0ifi �= jvi(a)ifi = jso the game designer defines f (v1,...,vn)=(k = arg max( vi)), wherepi=�0ifi �= kmaxj∈{1,...,n}\ivj(a)ifi = kwhere piconditionally equals 0 or the s econ d highes t valuation among players. Now, a mechanism isincentive compatible (IC) if ∀i∀vi: v�i�= vi.Iff(vi)=(a, p1,...,pn) and f(v�i,v−i)=(a�,p�1,...,p�n),then vi(a) − pi≥ vi(a�) − p�i. So,Theorem 9.2. A Vickery auction is IC.Why is this result bene fic ial ? Why not design a game which allows playe rs to behave naturally,i.e. li e ? It turns out there only IC game design mechanisms exist.Theorem 9.3. (Revelation Principle) If there is a mechanism, then it is a truthful mechanism .Proof. Suppose you design an ot he r mechanism. Because all utilities are private, each player willselect a strategy that maximizes her interests given the possible outcomes. Suppose that this iscomputed by an algorithm Mifor th e ith player. That is, the ith player inputs her true utility uito the algorithm, and it compu te s the best strategy, and submits then it to the mechanism G.Butthen, the game G together with the algorithms Miis an IC mechanism!Vickery’s auction works well for two reasons. First, it is in th e semilinear domain. Second, itmaximizes�ivi(a), which is the social welfare. Very few alternatives to Vickery auctions havebeen discover ed for different conditions.9.3 Vicker y -Cl a rke-Groves AuctionThe Vickery-Clarke-Groves (VCG) presents a general-form solution for IC mechanisms. The VCGdesigner defines a mechanism function, f (v1,...,vn)=(a, p1,...,pn), where a ∈ arg maxA�ivi(a),and defines the prices aspi(v1,...,vn)=hi(v−i) −�j�=ivj(a)Note, f or some a each player i is paid according to vi(a), the social welfare is�ivi(a), and somethingthe player cannot control (i.e. some tax suffered regardless of what that player declared). That isui= vi(a)+�j�=ivj(a) − hi(v−i),Lecture 9: Mechanism Design 9-3so VCG aligns the players’ and the design er ’ s interests. Since players want to assist the designermaximize social welfare, we haveTheorem 9.4. VCG is IC.Proof. Recall the definition of IC. Since vi(a)−pi≥ vi(a�)−p�i,then�jvj(a)−hi(v−i) ≥�jvj(a�)−hi(v−i), since we can add hi(v−i) to both s id es of the inequality, and a maximizes social welfare.9.4 Clarke Pivot RuleNow, we consider how to best defi ne hi. An uninteresting option i s to define hi= 0, but this intro-duces problems when the game payoff is large. Instead, consider the Clarke pivot rule (payment)hi(v−i) = maxb∈A�j�=ivj(b)where prices are defined aspi= maxb∈A�j�=ivj(b) −�j�=ivj(a)where pi= 0 if you are selected to ge t the item (absorb the cost). Otherwise, you earn valuedefined.Definition 9.1. Individually rational (IR) A mechanism is individually rational (IR) if allplayers receive nonnegative utility.Definition 9.2. No positive transfers (NPT) A mechanism has no positive trans f ers if noplayer is ever paid money.Because NTR an d IR, thenui=�ivi(a) −�j�=ivj(b) ≥
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