Christos H. Papadimitriou UC Berkeley www.cs.berkeley.edu/~christos Fun with Games Christos H. Papadimitriou UC Berkeley www.cs.berkeley.edu/~christos

(Mathematical tools: combinatorics, logic) Goal of CS Theory (1950-2000): Develop a mathematical understanding of the capabilities and limitations of the von Neumann computer and its software –the dominant and most novel computational artifacts of that time (Mathematical tools: combinatorics, logic) What should Theory’s goals be today? fun, May 29, 2001

fun, May 29, 2001

The Internet huge, growing, open, anarchic as information repository: huge, available, unstructured built, operated and used by a multitude of diverse economic interests theoretical understanding urgently needed fun, May 29, 2001

new math for the new Theory? cf: George Boole The Laws of Thought, 1854 Part I: propositional logic, Part II: probability cf: John von Neumann The Report on EDVAC, 1945 Theory of Games and Economic Behavior, 1944 (cf: Alan Turing On Computable Numbers, 1936 Studies in Quantum Mechanics, 1932-35) fun, May 29, 2001

Games, games… strategies strategies 3,-2 payoffs fun, May 29, 2001

matching pennies prisoner’s dilemma 1,-1 -1,1 3,3 0,4 4,0 1,1 auction chicken 1 … n 1 . n 0,0 0,1 1,0 -1,-1 0, v – y u – x, 0 fun, May 29, 2001

concepts of rationality undominated strategy (problem: too weak) dominating srategy (problem: too strong, rarely exists) Nash equilibrium (problem: may not exist) randomized Nash equilibrium (  P?) Theorem [Nash 1952]: Always exists. . fun, May 29, 2001

if a digraph with all in-degrees 1 has a source, then it must have a sink  Sperner’s Lemma  Brouwer’s Fixpoint Theorem  Kakutani’s Theorem Nash’s Theorem min-max theorem linear programming duality fun, May 29, 2001

More problems: Too many Nash equilibria Also, Nash equilibria may be “politically incorrect:” Prisoner’s dilemma Repeated prisoner’s dilemma? PD n Herb Simon (1969): Bounded Rationality “the assumption that reasoning and computation are infinitely cheap is often at the root of negative results in Economics” Idea: Prisoner’s dilemma played by memory-limited players? fun, May 29, 2001

c d d c d c d c c c, d c punish once tit-for-tat d d c d c,d c c d c c punish forever switch on d fun, May 29, 2001

e.g., pricing multicasts [Feigenbaum, P., Shenker, STOC2000] 52 30 costs {} 21 21 40 70 {11, 10, 9, 9} {14, 8} {9, 5, 5, 3} 32 {23, 17, 14, 9} {17, 10} utilities of agents in the node (u = the intrinsic value of the information to agent i, known only to agent i) i fun, May 29, 2001

We wish to design a protocol that will result in the computation of: x (= 0 or 1, will i get it?) v (how much will i pay? (0 if xi = 0) ) protocol must obey a set of desiderata: i i fun, May 29, 2001

strategy proofness: (w = u  x  v ) w (u …u …u )  w (u … u'…u ) 0  v  u, lim x = 1 strategy proofness: (w = u  x  v ) w (u …u …u )  w (u … u'…u ) welfare maximization  ui xi – c[T] = max marginal cost mechanism i i i u  i def i i i i i 1 i n i 1 i n budget balance  v = c ( T [x]) Shapley mechanism i i fun, May 29, 2001

our contribution: In the context of the Internet, there is another desideratum: Tractability: the protocol should require few (constant? logarithmic?) messages per link. This new requirement changes drastically the space of available solutions. fun, May 29, 2001

Shapley mechanism 0  v  u lim x = 1 strategy proofness: (w = u  x  v ) w (u …u …u )  w (u … u'…u ) welfare maximization  w = max marginal cost mechanism i i i u  i def i i i i i 1 i n i 1 i n budget balance  v = c ( T [x]) Shapley mechanism i i fun, May 29, 2001

Games: Fun Deep Useful The mathematical tool of choice of the new TCS fun, May 29, 2001

Bounding Nash equilibria: the price of anarchy cost of worst Nash equilibrium “socially optimum” cost s t 3/2 [Koutsoupias and P, 1998] general multicommodity network 2 [Roughgarden and Tardos, 2000] fun, May 29, 2001