Lecture #3: Game Theory and Social Networks Kyumars Sheykh Esmaili
Outline Games Modeling Network Traffic Using Game Theory
Games
Exam or Presentation Game You need to choose between exam or presentation: If you study exam, your grade will be 92, if don t study - 80; Presentation must be prepared jointly with a partner: if both you and your partner prepares, you ll get 100; if only one of you prepares, you ll get 92; if neither of you prepares, you ll get 84; You cannot contact your presentation partner, so you don t know what he does; Your final grade is the average of exam and presentation grades
Basic Ingredients of a Game A Game is any situation with the following three aspects: a set of players each player decides how to behave: chooses a strategy for each choice of strategies, each player receives a payoff that can depend on the strategies selected by everyone payoff matrix - payoffs for all players for all possible strategies Our interest is to understand how players will behave in a given game
Underlying Assumptions We focus on games with 2 players, one-shot games Assumptions: 1. A player maximizes her own payoff. Or a sum payoff of all players 2. Each player knows everything about a structure of a game 3. Rationality: each player chooses a strategy based on the beliefs about the strategies chosen by the other player; each player is experienced - doesn t do mistakes
Behavior in the Exam or Presentation Game if your partner studies for the exam, you get: 88 if you study for the exam, 86 if you prepare for the presentation if your partner prepares for the presentation, you get: 90 if you prepare for the presentation, 92 if you prepare for the exam
Strictly Dominant Strategy Preparing for the exam is a strictly dominant strategy for you Also for your partner, so we expect you both get 88 But! If you and your partner could somehow agree to prepare for the presentation, you would each be better off (90 each) And then: if your partner knew you are preparing for the presentation, he can choose to prepare for the exam, to be even better off (92)
The Prisoner s Dilemma Two suspects are the players, they choose between two possible strategies: Confess (C) or Not-Confess (NC) If Suspect 2 confess, then Suspect 1 gets: -4 if he confesses; -10 not confessing; If Suspect 2 doesn t confess, then Suspect 1 gets: 0 by confessing; -1 by not confessing; 9
Best Responses and Dominant Strategies S - strategy chosen by Player 1, T - strategy chosen by Player 2 there is an entry in the payoff matrix (S, T) P1(S,T) - payoff to Player 1, P2(S, T) - payoff to the Player 2 S for Player 1 is a best response to a strategy T for Player 2 if: P1(S,T)>=P1(S,T) S is a strict best response for T if: P1(S,T)>P1(S,T)
Dominant Strategy Dominant strategy for Player 1 is a strategy that is a best response to every strategy of Player 2 Strictly dominant strategy for Player 1 is a strategy that is a strict best response to every strategy of Player 2 In Prisoner s Dilemma both players had strictly dominant strategies, so it was easy to reason what was likely to happen
Games with Only One Player with a Strictly Dominant Strategy Firm 1 and Firm 2 choose between producing low-priced product or upscale product Market is divided into 2 segments: low-priced version of a product (60%) upscale version (40%) Firm 1 is more popular, so it gets 80% of market if it competes with Firm 2 (20%) Firm 1 has a strictly dominant strategy Low-priced 12
Games with no Player with a Strictly Dominant Strategy Three-Client Game: (2 firms, 3 clients: A, B and C) if both firms approach the same client => client gives half of its business firm 1 is too small, if it approaches a client alone - it gets 0 A is a large client => it is only possible to approach together if firm 2 approaches B or C => it gets full business A costs 8, B or C cost 2 No strictly dominant strategy - how should we reason about the outcome? 13
Nash Equilibrium John Nash proposed in 1950 the concept (later received Nobel Prize in Economics) Suppose, Player 1 chooses a strategy S and Player 2 chooses a strategy T. Then (S,T) is a Nash Equilibrium if S is a best response for T and T is a best response for S Idea: if players choose strategies that are best responses to each other, then no player has an incentive to change to alternative strategy 14
Nash Equilibrium (A,A) - Nash Equilibrium (A,A) - is the only Nash Equilibrium in this game (check (B,C) for example ) 15
Multiple Equilibria: Coordination Games Some natural games can have more than one Nash equilibrium Coordination game: two players share a goal to coordinate on the same strategy (PowerPoint,PowerPoint) and (Keynote,Keynote) - 2 Nash equilibria 16
Variants on the Basic Coordination Game Unbalanced coordination game we can predict that when the players have to choose, they will select strategies so as to reach the equilibrium that gives higher payoffs to both of them If you and your partner don t agree on which software to choose
Basic Coordination Game: Stag Hunt Game Two people are hunting: if they hunt together, they can catch a stag on their own they can catch a hare only
Anti-Coordination Game: The Hawk-Dove Game if both behave passively - they divide the food evenly, each gets a payoff 3 if one behaves aggressively while the other behaves passively - they get 5:1 if both behave aggressively, they destroy the food - get 0
Anti-Coordination Game: Variant of Exam or Presentation? Modification to the Exam or Presentation : if none of you prepares for the presentation - you get very low grade - 60 Two equilibria: (Presentation, Exam) and (Exam, Presentation). One of you must behave passively and prepare for the presentation You want to prepare for the exam, but risk to be penalized
Summary, so far What s in a Game? Prisoner s Dilemma: strictly dominant strategies do not give the best outcome for players Nash Equilibrium: stable state in a game when both players use strategies that are best responses to each other Multiple Equilibria: if you attempt to go for a higher payoff equilibrium - you risk get even lower one
Mixed Strategies In games where there is no Nash equilibria at all if players are allowed to behave randomly => equilibria always exist Example of a game without equilibria: Attack-Defense Game Player1 - attacker; Player2 defender Attacker can use strategies A or B. Defender can defend against A or B. If the Defender defends against the actual attack, then he gets higher payoff; If the Defender defends against the wrong attack, then the Attacker gets the higher payoff 22
Matching Pennies: An Example of Attack- Defense Game Two people hold a penni and simultaneously choose whether to show heads (H) or tails (T) Player1 loses his penni if they match Player2 wins a penni if they don t match Zero-sum game - payoffs sum to zero in every outcome; In zero-sum games players interest are in direct conflict 23
Another Example: Run/Pass Game In many sport (i.e. Football, Basketball, )
Another Example: Penalty-Kick Game Kicker Options Kick the ball to left or right Goalie Options Jump to the left or right
Randomization of Strategies In Matching Pennies, no two strategies that are best responses to each other => no Nash equilibrium In real life, players try to hide which strategy they are choosing We will model this situation as randomizing players behavior between strategies H and T
Mixed Strategies Randomization - players are choosing a probability with which they will play H or T Player1 commits to play H with probability p; and to play T with probability 1-p Player2 commits to play H with probability q; and to play T with probability 1-q Now we have a set of strategies (instead of only two) corresponding to the interval of numbers between 0 and 1 Committing to definitely play H or T (selecting probabilities of 1 or 0) - pure strategies
Payoffs from Mixed Strategies Suppose, Player2 commits to play H with probability q, and T with probability 1-q Then, if Player1 chooses pure strategy H => receives -1 with probability q and +1 with probability 1-q Expected value of the payoff for Player1: (-1)q+(1)(1-q) = 1-2q for Player2: (1)q+(-1)(1-q) = 2q - 1 28
Equilibrium with Mixed Strategies No pure strategy can be part of Nash equilibrium 1-2q = 2q-1, otherwise one of the pure strategies H or T is the unique best response 1-2q=2q-1 => q=1/2. Similar, p=1/2. q=p=1/2 is the only Nash equilibrium for this game
Indifference The choice of q=1/2 by Player 2 makes Player 1 indifferent between playing H or T Each player wants their behavior to be unpredictable to the other, so that their behavior can t be taken advantage of Interpretations of mixed-strategy equilibria: tennis - player randomly decides where to serve the ball playing cards - randomly deciding whether to bluff or not matching pennies - it is enough for you to expect that when you meet an arbitrary person, they will play with probability 1/2
Offense Probabilities: Run/Pass Game Pass: p Run: 1 - p Defense probabilities Defend Pass: q Defend Run: 1 - q 31
Penalty-Kick Game Kicker Probabilities: Left: 0.39 Right: 0.61 Goalie probabilities Left: 0.42 Right: 0.58 32
Pareto Optimality In Nash equilibrium - players are optimizing individually But what if we want an outcome that is good for society? Pareto Optimality: A choice of strategies is Pareto-optimal if no other choice of strategies in which all players receive payoffs at least as high, and at least one player receives a strictly higher payoff Sometimes, reaching PO is not possible without a binding agreement between the players 33
Social Optimality A choice of strategies is a social welfare maximizer (or socially optimal) if it maximizes the sum of the players payoffs Outcomes that are socially optimal must be Pareto-optimal Pareto-optimal is not necessarily social optimal
Modeling Network Traffic Using Game Theory 35
Traffic at Equilibrium Transportation network as a directed graph Suppose, we have 4000 cars All cars take upper route: 85 minutes Divided evenly: 65 minutes Nash equilibrium - is when the cars divide up evenly: with even balance between two routes, no driver has an incentive to switch over to the other route 36
Braess s Paradox A small change to the network can lead to a counterintuitive situation Adding CD - a fast highway (0-minutes to drive) With CD, there is a unique Nash equilibrium Worse travel time for everyone: 80 minutes This phenomenon is Braess s Paradox: adding resources to a transportation network can sometimes hurt performance at equilibrium 37
Readings: Chapters 6 and 8 of the textbook