Computing Best-Response Strategies in Infinite Games of Incomplete Information

Daniel Reeves and Michael Wellman University of Michigan

Definitions

Infinite Game

= infinite action spaces

Incomplete Information

private to the players

Type

= payoffs depend on information that is = a player’s private information

One-shot Game

= players each choose a single action simultaneously and then immediately receive a payoff

Strategy

= a mapping from type to action

Best-Response Strategy

= optimal strategy given known strategies of the other players

Nash Equilibrium

= profile of strategies such that each strategy is a best response to the others

Bayes-Nash Equilibrium

= generalization of NE to the case of incomplete information, for expected-utility maximizing players

Finite Game Approximations

Finite game solvers:  Gambit  Gala  Gametracer Why not discretize?

 Introduces qualitative differences  Computationally intractable

Our Class of Games

2-player, one-shot, infinite games of incomplete information Piecewise uniform type distributions Payoff functions of the form:

Games in our Class

Other games: War of Attrition, Incomplete info versions of Cournot and Bertrand games

Piecewise Linear Strategies

Specified by the vectors

c

,

m

,

b

Existence and Computation of Piecewise Linear Best Responses Theorem 1: Given a payoff function with I regions as above, an opponent type distribution with cdf F that is piecewise uniform with J pieces, and a piecewise linear strategy function with K pieces, the best response is itself a piecewise linear function with no more than 2(I-1)(J+K-2) piece boundaries.

The Proof

For arbitrary own type t, and opponent type a random variable T, find own action a maximizing E T [u(t,a,T,s(T))] (Numerical maximization not applicable due to parameter t) Above works out to be a piecewise polynomial in a (parameterized by t) For given t, finding optimal a is straightforward Remains to find partitioning of type space such that within each type range, optimal action is a linear function of t This can be done in polynomial time

Example: First-Price Sealed Bid Auction (FPSB) Types (valuations) drawn from U[0,1] Payoff function: Known Bayes-Nash equilibrium (McAfee & McMillan, 1987): a(t)=t/2 Found in as few as one iteration from a variety of seed strategies

Example: Supply-chain Game

Producers’ Costs U[0,1] Consumer’s Valuation v in [1.5,3] (known) Payoff function: bid-cost if bid+bid2 <= v 0 otherwise Producer 1 Producer 2 Consumer

Proving a Bayes-Nash Equilibrium

Candidate Strategy: 2/3 v – 1/2 if cost < 2/3 v – 1 cost/2 + v/3 otherwise Compute best response…

Computing Best Response

Expected payoff , EP(b) =(b-c)*p(b+b2<=v) =(b-c)*[p(c2<=2/3v-1)*p(b+2/3v-1/2<=v | c2<=2/3v-1) +p(c2>2/3v-1) * p(b+c2/2+v/3<=v | c2 > 2/3v-1)] =(b-c)*[(2/3v-1)*p(b<=v/3+1/2) +p(2/3v-1 < c2 < 4/3v-2b)] Case 1: b<=2/3v-1/2 EP(b) = (b-c)*[(2/3v-1)*1 + (2-2/3v)] = (b-c) ==> b* = 2/3v-1/2 ==> EP1(b*) = 2/3v-1/2-c Case 2: 2/3v-1/2 < b < v/3+1/2 EP(b) = (b-c)*[(2/3v-1)+(2/3v-2b+1)] = (b-c)*(4/3v-2b) ==> b* = c/2+v/3 ==> EP2(b*) = (3c-2v)^2/18 Case 3: b > v/3+1/2 ==> EP3(b) = 0

Computing Best Response (2)

EP1(b*) > EP2(b*) iff c < 2/3 v – 1 Therefore, best-response is… 2/3 v – 1/2 if c < 2/3 v – 1 c/2 + v/3 otherwise

Example: Bargaining Game

(aka, sealed-bid k-double auction) Buyer and seller place bids, transaction happens iff they overlap Transaction price is some linear combination of the bids Known equilibrium (Chatterjee & Samuelson, 1983) for seller (1) and buyer (2): Found in several iterations from truthful bidding

Provision Point Mechanism

(aka, Public Good or Voluntary Participation game) 2 agents want to jointly acquire a good costing C Mechanism: simultaneously offer contributions; buy iff sum > C and split the excess (C – sum) evenly Nash: 2/3 t + C/4 – 1/6

Shared-Good Auction

New mechanism, similar to the divorce settlement game; undoes provision point Agents place bids for a good they currently share, valuations ~U[A,B] High bidder gets the good and pays half its bid to the low bidder in compensation

Equilibrium in Shared-Good Auction Found in one iteration from truthful bidding (for any specific [A,B])

Vicious Vickrey Auction

Generalization of a Vickrey Auction (Brandt & Weiss, 2001) to allow for disutility from opponent’s utility (eg, business competitors) Brandt & Weiss consider only the complete information version

Equilibrium in Vicious Vickrey

a(t) = (k+t)/(k+1) Reduces to truthful bidding for the standard Vickrey Auction (k=0) Iterated best-response solver finds this equilibrium (for specific values of k) within several iterations from a variety of seed strategies

Conclusions

First algorithm for finding best-response strategies in a broad class of infinite games of incomplete information Confirms known equilibria (eg, FPSB), confirms equilibria we derive here (Supply Chain game), discovers new equilibria (Shared-good auction, Vicious Vickrey) Goal: characterize the class of games for which iterated best-response converges