1 What is Game Theory About? r Analysis of situations where conflict of interests is present r Goal is to prescribe how conflicts can be resolved 2 2 r Game of Chicken m driver who steers away looses r What should drivers do?

2 Applications of Game Theory r Theory developed mainly by mathematicians and economists r Widely applied in many disciplines m from economics to philosophy, including computer science (Systems, Theory and AI) m goal is often to understand some phenomena r “Recently” (>20 years) applied to communication networks m wider interest starting around 2000

3 Limitations of Game Theory r No unified solution to general conflict resolution r Real-world conflicts are complex m models can at best capture important aspects r Players are (usually) considered rational m determine what is best for them given that others are doing the same r No unique prescription m not clear what players should do r But it can provide intuitions, suggestions and partial prescriptions m best mathematical tool we currently have

4 What is a Game? r A Game consists of m at least two players m a set of strategies for each player m a preference relation over possible outcomes r Player is general entity m individual, company, nation, protocol, animal, etc r Strategies m actions which a player chooses to follow r Outcome m determined by mutual choice of strategies r Preference relation m modeled as utility (payoff) over set of outcomes

5 Classification of Games r Many, many types of games m two major categories r Non-Cooperative (Competitive) Games m individualized play, no bindings among players r Cooperative Games m play as a group, possible bindings

6 Matrix Game (Normal form) r Simultaneous play m players analyze the game and write their strategy on a paper r Combination of strategies determines payoff ABC A(2, 2)(0, 0)(-2, -1) B(-5, 1)(3, 4)(3, -1) Player 1 Player 2 Strategy set for Player 1 Strategy set for Player 2 Payoff to Player 1 Payoff to Player 2 r Representation of a game

7 More Formal Game Definition r Normal form (strategic) game  a finite set N of players m a set strategies for each player m payoff function for each player where is the set of strategies chosen by all players  A is the set of all possible outcomes r is a set of strategies chosen by players m defines an outcome r

8 Two-person Zero-sum Games r One of the first games studied m best understood type of game r Players interests are strictly opposed m what one player gains the other loses m game matrix has single entry (gain to player 1) r Intuitive solution concept m players maximize gains m unique solution

9 Analyzing the Game r Player 1 maximizes matrix entry, while player 2 minimizes ABCD A1210 B C5243 D-1612 Player 1 Player 2 Strictly dominated strategy (dominated by C) Strictly dominated strategy (dominated by B)

10 Dominance r Strategy S strictly dominates a strategy T if every possible outcome when S is chosen is better than the corresponding outcome when T is chosen r Dominance Principle m rational players never choose strictly dominated strategies r Idea: Solve the game by eliminating strictly dominated strategies! m iterated removal

11 Solving the Game LMR T-24 B323 Player 1 Player 2 r Iterated removal of strictly dominated strategies m Player 1 cannot remove any strategy (neither T or B dominates the other) m Player 2 can remove strategy R (dominated by M) m Player 1 can remove strategy T (dominated by B) m Player 2 can remove strategy L (dominated by M) m Solution: P 1 -> B, P 2 -> M payoff of 2

12 Solving the Game ABD A04 C423 D510 Player 1 Player 2 r Removal of strictly dominated strategies does not always work r Consider the game r Neither player has dominated strategies r Requires another solution concept

13 Analyzing the Game ABD A04 C423 D510 Player 1 Player 2 Outcome (C, B) seems “stable” m saddle point of game

14 Saddle Points r An outcome is a saddle point if it is both less than or equal to any value in its row and greater than or equal to any value in its column r Saddle Point Principle m Players should choose outcomes that are saddle points of the game r Value of the game m value of saddle point outcome if it exists

15 Why Play Saddle Points? r If player 1 believes player 2 will play B m player 1 should play best response to B (which is C) r If player 2 believes player 1 will play C m player 2 should play best response to C (which is B) ABD A04 C423 D510 Player 1 Player 2

16 Why Play Saddle Points? ABD A04 C423 D510 Player 1 Player 2 r Why should player 1 believe player 2 will play B? m playing B guarantees player 2 loses at most v (which is 2) r Why should player 2 believe player 1 will play C? m playing C guarantees player 1 wins at least v (which is 2) Powerful arguments to play saddle point!

17 Solving the Game (min-max algorithm) r choose minimum entry in each row r choose the maximum among these r this is maximin value ABCD A4325 B-1020 C7513 D Player 1 Player r choose maximum entry in each column r choose the minimum among these r this is the minimax value r if minimax == maximin, then this is the games’ saddle point

18 Multiple Saddle Points ABCD A3225 B2-100 C5223 D Player 1 Player r In general, game can have multiple saddle points r Same payoff in every saddle point m unique value of the game r Strategies are interchangeable m Example: strategies (A, B) and (C, C) are saddle points then (A, C) and (C, B) are also saddle points

19 Games With no Saddle Points r What should players do? m resort to randomness to select strategies ABC A20 B-531 Player 1 Player 2

20 Mixed Strategies r Each player associates a probability distribution over its set of strategies m players decide on which prob. distribution to use r Payoffs are computed as expectations CD A40 B-53 Player 1 1/32/3 Payoff to P1 when playing A = 1/3(4) + 2/3(0) = 4/3 Payoff to P1 when playing B = 1/3(-5) + 2/3(3) = 1/3 r How should players choose prob. distribution?

21 Mixed Strategies r Idea: use a prob. distribution that cannot be exploited by other player m payoff should be equal independent of the choice of strategy of other player m guarantees minimum gain (maximum loss) CD A40 B-53 Player 1 Payoff to P1 when playing A = x(4) + (1-x)(0) = 4x Payoff to P1 when playing B = x(-5) + (1-x)(3) = 3 – 8x 4x = 3 – 8x, thus x = 1/4 r How should Player 2 play? x(1-x)

22 Mixed Strategies r Player 2 mixed strategy m 1/4 C, 3/4 D m maximizes its loss independent of P1 choices r Player 1 has same reasoning CD A40 B-53 Player 1 Payoff to P2 when playing C = x(-4) + (1-x)(5) = 5 - 9x Payoff to P2 when playing D = x(0) + (1-x)(-3) = x 5 – 9x = x, thus x = 2/3 Player 2 x (1-x) Payoff to P2 = -1

23 Minimax Theorem r Every two-person zero-sum game has a solution in mixed (and sometimes pure) strategies m solution payoff is the value of the game m maximin = v = minimax m v is unique r Proved by John von Neumann in 1928! m birth of game theory…

24 Two-person Non-zero Sum Games r Players are not strictly opposed m payoff sum is non-zero AB A3, 42, 0 B5, 1-1, 2 Player 1 Player 2 r Situations where interest is not directly opposed m players could cooperate

25 What is the Solution? r Ideas of zero-sum game: saddle points r pure strategy equilibrium r mixed strategies equilibrium m no pure strategy eq. AB A5, 0-1, 4 B3, 22, 1 Player 1 Player 2 AB A5, 42, 0 B3, 1-1, 2 Player 1 Player 2

26 Multiple Solution Problem r Games can have multiple equilibria m not equivalent: payoff is different m not interchangeable: playing an equilibrium strategy does not lead to equilibrium AB A1, 41, 1 B0, 12, 2 Player 1 Player 2 equilibria

27 The Good News: Nash’s Theorem r Every two person game has at least one equilibrium in either pure or mixed strategies r Proved by Nash in 1950 using fixed point theorem m generalized to N person game r Def: An outcome o* of a game is a NEP (Nash equilibrium point) if no player can unilaterally change its strategy and increase its payoff r Cor: any saddle point is also a NEP

28 The Prisoner’s Dilemma r One of the most studied and used games m proposed in 1950s r Two suspects arrested for joint crime m each suspect when interrogated separately, has option to confess or remain silent SC S2, 210, 1 C1, 105, 5 Suspect 1 Suspect 2 payoff is years in jail (smaller is better) single NEP better outcome

29 Pareto Optimal r Prisoner’s dilemma: individual rationality SC S2, 210, 1 C1, 105, 5 Suspect 1 Suspect 2 r Another type of solution: group rationality m Pareto optimal r Def: outcome o* is Pareto Optimal if no other outcome is better for all players Pareto Optimal

30 Game of Chicken Revisited 2 2 r Game of Chicken (aka. Hawk-Dove Game) m driver who swerves looses swervestay swerve0, 0-1, 5 stay5, -1-10, -10 Driver 1 Driver 2 Drivers want to do opposite of one another Will prior communication help?