Presentation Reiknirit, rökfræði og reiknanleiki - 08.73.11 Anna Ólafsdóttir Björnsson

1. Algorithmic Game Theory - Basic Solution Concepts and Computational IssuesÉva Tardos and Vijay V. Vazirani Presentation Reiknirit, rökfræði og reiknanleiki - 08.73.11 Anna Ólafsdóttir Björnsson

2. 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 Anna Ólafsdóttir Björnsson

3. 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) Anna Ólafsdóttir Björnsson

4. 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 Anna Ólafsdóttir Björnsson

5. 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 Anna Ólafsdóttir Björnsson

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

7. Games, Strategies, Cost, Payoffs • Simultaneous Move Game: • Each player i selects game strategy defined in a vector of strategies si∈ Si • Vector of strategies of all players: • s = (s1 ... Sn) • Set of all possibities to pick strategy: • S = xiSi • Assign values for each move for a player • Utilities ui: S-R and Costs ci: S-R • Costs and payoffs can be used interchangeably since ui(s) = -ci(s) Anna Ólafsdóttir Björnsson

8. 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 Björnsson

9. 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” Anna Ólafsdóttir Björnsson

10. 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 Björnsson

11. 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 Björnsson

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

13. 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 Anna Ólafsdóttir Björnsson

14. 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 Anna Ólafsdóttir Björnsson

15. Different types of cooperative games Anna Ólafsdóttir Björnsson

16. 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 Björnsson

17. 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. (www.wikipedia.org) Anna Ólafsdóttir Björnsson