Algorithmic Game Theory - Basic Solution Concepts and Computational Issues Éva Tardos and Vijay V. Vazirani Presentation Reiknirit, rökfræði og reiknanleiki Anna Ólafsdóttir Björnsson

Overview  Introduction to basic game- theoretic definitions and tools  Different methods are suitable for different games  Pros and cons of some of the different methods introduced 2Anna Ólafsdóttir Björnsson

The questions the paper deals with  Game theory methods  Games, Strategies, Cost and Payoffs  Introducing basic solution concepts  How easy is it to find an equilibrium  Does "natural game play” lead the players to an equilibrium?  Examples (Nash and others) 3Anna Ólafsdóttir Björnsson

Game theory methods  There are several ways of presenting game theory  For each game may be a different preference  Cost matrices show what each player gains/loses in each move  Equilibriums are know for several games 4Anna Ólafsdóttir Björnsson

Example: Prisoner’s Dilemma  A classic and well studied game is the Prisoner’s Dilemma  2 prisoners can a) confess b) remain silent  Each can reduce own sentence by confessing but his/her confession is only rewarded if the other one chooses to remain silent 5Anna Ólafsdóttir Björnsson

Cost matrix for Prisoner’s Dilemma P2 P1 ConfessSilent Confess Silent 6Anna Ólafsdóttir Björnsson

Games, Strategies, Cost, Payoffs Simultaneous Move Game:  Each player i selects game strategy defined in a vector of strategies s i ∈ S i  Vector of strategies of all players: s = (s 1... S n )  Set of all possibities to pick strategy: S = x i S i  Assign values for each move for a player  Utilities u i : S-R and Costs c i : S-R  Costs and payoffs can be used interchangeably since u i (s) = -c i (s) 7Anna Ólafsdóttir Björnsson

Basic solution concepts  What affects the strategy chosen: Unique best strategy or not? Individual choices – often low payoff  Example: Prisoner’s Dilemma Strategy chosen by one player effects the other – but individually chosen Willingness to risk for a better payoff A 3rd party deciding for both players  Example: Traffic lights Anna Ólafsdóttir Björnsson8

Basic solution concepts (cont.) Games that have unique best strategy – "Dominant strategy solution”  Prisoners Dilemma, Pollution Game etc.  Solution that may not give optimal payoff to any of the players  Designing such games. Example: Auction and Second price auction-Vickery auction Where the highest bidder pays 2nd highest price and each player independant of others ... but "games rarely possess dominent strategy solution” 9Anna Ólafsdóttir Björnsson

Basic solution concepts (cont.) More common than using unique best strategy:  Player tries to maximize the payoff  Strategy where no single player can individually improve his/her welfare Players cannot change their strategy Get better results not changing  With and without use of Nash Equilibria  Unique best strategy is a Nash Equilibrium ...but not all Nash Equilibria are unique  Tragedy of the commons, Battle of sexes Anna Ólafsdóttir Björnsson10

Basic solution concepts (cont.) Any game with a finite set of players and finit set of strategies has a Nash equilibrium of mixed strategies Anna Ólafsdóttir Björnsson11

How easy to find an equilibrium?  Correlated equilibria – can be found in polynominal time  Two persons zero-sum lineral computing lead to Nash (PPAD) 12Anna Ólafsdóttir Björnsson

Natural game play and equilibrium Does "natural game play” lead the players to an equilibrium? Assume that most natural "game playing” strategy is "best response” Learning strategies can improve results Not all types of games lead to Nash equilibria but instead to correlated equilibria if "learning strategies” are used 13Anna Ólafsdóttir Björnsson

More types of games Games with multiple turns of moves can be reduced to simultaneous move games and thus to Nash equilibria Without full information - Bayesian Games – use probability distribution Cooperative games – players coordinate their actions 14Anna Ólafsdóttir Björnsson

Different types of cooperative games NameDescriptionMethods“Games” Strong Nash Equilibrium No subset A of players have a way to simultaneously change their strategies, improving each of the participants welfare s A vector of players startegies s -A vector of strategies for players outside A u i (s A, s -A ) but NOT u i (s) ≤ u i (s’ A, s -A ) Drawback - applies to few games: Stable marriage problem Fair division and cost- sharing Transferable utility games Game dividing some value or sharing cost between participants fairly. Cost-sharing is in the core if no subset of players would decrease their shares by breaking away from the whole set N = all players A = subset of players associated with a cost c(A), c(N), v(A) cost of serving players Cost share vector in core if: ∑ i ∈A x i ≤ c(A) Shapley value - Order player set N 1,... N marginal cost of player i is c(N i )- c(N i-1 ) 15Anna Ólafsdóttir Björnsson

Market “games” and their algorithimc issues Mostly non-algorithmic theory - need to find (more) algorithms Lemma: A uniform price of x on all goods is feasible iff all goods can be sold in such a way that each buyer gets goods that she is interested in Lemma: The value x* is feasible for the problem restrict to goods in A-S* and buyers in B- Γ(S*). Furthermore, in the subgraph of G induced on A-S* and B- Γ(S*), all the vertices have nonzero degree. There exists an algorithm that computes equilibrium prices and allocations in polynominal time (pg.25). Anna Ólafsdóttir Björnsson16

References  Éva Tardos and Vijay V. Vazirani: Basic Solution Concepts and Computational Issues. Algorithmic Game Theory  Algorithic Game Theory. Some other articles as a side material.  Wikipedia. Game Theory. ( 17Anna Ólafsdóttir Björnsson