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© 2009 Pearson Education Canada 15/1 Chapter 15 Introduction to Game Theory
© 2009 Pearson Education Canada 15/2 Game Theory: Basic Definitions Entities like firms, individuals, or governments that make choices in the game are referred to as players. The actions players choose – for a firm things such as output and/or price; for a soldier such things as run, hide or shoot, are called strategies.
© 2009 Pearson Education Canada 15/3 Game Theory: Basic Definitions The list of strategies a player can choose from is the player’s strategy set (run, hide, shoot). A strategy combination is just a single combination of the various player’s strategies. For two players, an example would be: (shoot, shoot). The set of strategy combinations is a list of all possible strategies that can result in a game.
© 2009 Pearson Education Canada 15/4 Game Theory: Basic Definitions A payoff is the outcome of a given strategy. If each player’s payoff is determined not only by their own strategy, but also by other players’ strategies, the game is said to have strategic interdependence.
© 2009 Pearson Education Canada 15/5 Game Theory: Basic Definitions A player’s best response to the given strategies of other players is known as a best response function. If every player’s strategy is a best response to the strategies of all other players, the combination is known as an equilibrium strategy combination.
© 2009 Pearson Education Canada 15/6 Game Theory: Basic Definitions Cournot-Nash equilibrium - An equilibrium strategy combination where there is nothing any individual player can independently do that increases that player’s payoff. Each player’s own strategy maximizes that player’s own payoff.
© 2009 Pearson Education Canada 15/7 Game Theory: Basic Definitions Normal forms simply represent the outcomes in payoff matrix (connects the outcomes in an obvious way). Extensive forms are a game tree. Each decision point (node) has a number of branches stemming from it; each one indicating a specific decision. At the end of the branch, there is another node or a payoff.
© 2009 Pearson Education Canada 15/8 Game Theory: Basic Definitions A strategy better than all others, regardless of the actions of others, is a dominant strategy. If one strategy is worse than another for some players, regardless of the actions of other players, it is a dominated strategy.
© 2009 Pearson Education Canada 15/9 Game theory is based on the following modelling assumptions: There are a few producers (players) in the industry (game). Each player chooses an output or pricing strategy. Each strategy produces a result (payoff) for that player. The payoff for each player is dependent upon the strategy he/she selects and that selected by other players.
© 2009 Pearson Education Canada 15/10 Sharing: Game of Put and Keep Janet decides to play a game with her two children Jay and Jill. Each child is to go to his/her room where they have no way of communicating with each other. Janet gives each child $9 and the option to PUT (P) the money back or KEEP (K) it.
© 2009 Pearson Education Canada 15/11 Sharing: Game of Put and Keep Then Janet collects the two envelopes and adds $6 whenever she finds $9. Once this is done, Janet dumps the contents of the envelopes onto the kitchen table and splits the money evenly between Jay and Jill. The payoff matrix is shown in Figure 15.1
© 2009 Pearson Education Canada 15/12 Figure 15.1 The payoff matrix for Jill and Jay
© 2009 Pearson Education Canada 15/13 Figure 15.2 The payoff matrix for Jill and Jay with preference arrows.
© 2009 Pearson Education Canada 15/14 Sharing: Game of Put and Keep Each child has a dominant strategy to keep the money. Thus the dominant equilibrium strategy combination is (K,K) with payoff (9,9). Janet’s plan to get her children to cooperate does not work.
© 2009 Pearson Education Canada 15/15 Janet’s Second Attempt Like most mothers, Janet is not deterred by one failure. She creates a different game (the same as the first), but now she tells the children if they keep the money, they have to bring it to the table and throw it in with whatever comes out of the envelope. The payoff matrix is shown in Figure 15.3.
© 2009 Pearson Education Canada 15/16 Figure 15.3 The share everything game
© 2009 Pearson Education Canada 15/17 Janet’s Second Attempt It is obvious that both players again have a dominant strategy, which is to put the money back in the envelope (PP). This combination is Pareto-optimal because there is no other strategy combination that makes at least one player better and does not make the other worse off.
© 2009 Pearson Education Canada 15/18 Janet’s Second Attempt Notice that in the first game, the equilibrium was not Pareto-optimal. Had Jay and Jill placed the money back in the envelope, they both would have been better off by $6. An equilibrium can exist that is not Pareto-optimal.
© 2009 Pearson Education Canada 15/19 Figure 15.4 The prisoner’s dilemma
© 2009 Pearson Education Canada 15/20 The Prisoner’s Dilemma Figure 15.4 shows payoffs for the two individuals suspected of car theft. The figures represent the jail time in months for Petra and Ryan. What is the equilibrium outcome of this game?
© 2009 Pearson Education Canada 15/21 From Figure 15.4 An easy way to find equilibrium is to draw arrows showing the direction of strategy preferences for each player. Horizontal arrows show preferences of player 2, vertical arrows show preferences for player 1. Where the two arrows meet, there is a Nash equilibrium (see Figure 15.5).
© 2009 Pearson Education Canada 15/22 Figure 15.5 Nash equilibrium in the PD game
© 2009 Pearson Education Canada 15/23 From Figure 15.5 The arrows meet where both Petra and Ryan fink (Fink, Fink) and this is the equilibrium for the game. –Note that this is a one shot game with no consequences apart from the given payoffs. –The equilibrium results form a dominant strategy for both players. –The equilibrium outcome is not Pareto- Optimal (both would be better off if they both remained silent).
© 2009 Pearson Education Canada 15/24 Nash Equilibrium In the next section on coordinated games, there is no dominant strategy, hence the interaction among players is more meaningful and the Nash equilibrium more useful.
© 2009 Pearson Education Canada 15/25 Nash Equilibrium Strategy combinations (S 1 *, S 2 *, … S N * ) is a Nash equilibrium strategy combination if every player’s strategy is a best response to the collection of strategies of the remaining players. To find the Nash equilibrium, first compute the best responses of all players and intersect them.
© 2009 Pearson Education Canada 15/26 Coordination Games Often situations may have no equilibrium or they may have multiple equilibria. This is said to be a coordination problem. A game of coordination has multiple Nash equilibria and requires some mechanism for determining the final outcome.
© 2009 Pearson Education Canada 15/27 Coordination Games: An Example Figure 15.8 shows the payoffs for various strategies using Microsoft Word (Dean’s preference) and Corel’s WordPerfect (Richard’s favourite). The figures represent how much better/worse each author is under the various strategies measured in more/less papers written.
© 2009 Pearson Education Canada 15/28 Figure 15.8 Choosing a word processor
© 2009 Pearson Education Canada 15/29 From Figure 15.8 As indicated by the arrows, there are two equilibria in this game. Therefore the Nash equilibrium is insufficient to identify the actual outcome. There exists a coordination problem when the players must decide on what equilibrium to settle on.
© 2009 Pearson Education Canada 15/30 The Game of Chicken and the Word Processor Game Perhaps our two academic writers try to solve their dispute over which software to use with a game of chicken: The person who swerves first must use the other’s preferred software. (Figure 15.9). Unfortunately, it is also a game with two equilibria and requires some outside coordination as well.
© 2009 Pearson Education Canada 15/31 Nash equilibrium in word processor game
© 2009 Pearson Education Canada 15/32 How Do the Players Decide a Strategy in Coordination Games? There is no definitive method of solving coordination games, actual outcomes often depend upon: laws, social customs or pre-emptive moves by players before the game. In some cases there simply is no equilibrium.
© 2009 Pearson Education Canada 15/33 Games of Plain Substitutes and Plain Complements Games in which each player’s payoff diminishes as the values of the other player’s strategy increases are known as games of plain substitutes. In games of plain substitutes, the players impose negative externalities on each other.
© 2009 Pearson Education Canada 15/34 Games of Plain Substitutes and Plain Complements Games in which each player’s payoff increases as the values of the other player’s strategy increases are known as games of plain complements. In games of plain complements, the players impose positive externalities on each other.
© 2009 Pearson Education Canada 15/35 Games of Plain Substitutes with Simultaneous Moves The cross-effects in the payoff functions are negative. There exists mutual negative externalities. y 1 0 and y 2 0 are the Nash equilibrium values of the strategies. From the Nash equilibrium, y 1 0 is a best response to y 2 0
© 2009 Pearson Education Canada 15/36 Figure 15.10 Nash equilibrium for a game of plain substitutes
© 2009 Pearson Education Canada 15/37 Games of Plain Substitutes with Simultaneous Moves (continued) Y 1 0 solves the constrained maximization problem: Maximize by choice of y 1 and y 2 п 1 (y 1, y 2 ) subject to y 2 = y 2 * Indifference curve п 1 (y 1, y 2 ) is tangent to the constraint at the Nash equilibrium (y 1 *, y 2 * ) in Figure 15.10. Because п 1 (y 1, y 2 ) decreases as y 2 increases, this indifference curve must lie below the line y 2 = y 2 * elsewhere.
© 2009 Pearson Education Canada 15/38 Games of Plain Substitutes with Simultaneous Moves For the same reason, the set of strategy combinations that One prefers to the Nash equilibrium lies below this indifference curve, as indicated by the downward–pointing arrows in Figure 15.10.
© 2009 Pearson Education Canada 15/39 Games of Plain Substitutes with Simultaneous Moves For Two’s indifference curve through the Nash equilibrium. It must be tangent to the line y 1 = y 1 at (y 1 *, y 2 * ). Elsewhere it must lie to the left of the line y 1 = y 1 * and the set of strategy combinations. Two’s preferences to the Nash equilibrium lie to the left of this indifference curve.
© 2009 Pearson Education Canada 15/40 From Figure 15.10 All strategy combinations in the Lense of Missed Opportunity are preferred by both players to the Nash equilibrium. Notice that in the Nash equilibrium, the players choose strategy values that are in some sense “too large” and a strategy combination that Pareto-dominates the Nash equilibrium, has smaller values for both strategies.
© 2009 Pearson Education Canada 15/41 Figure 15.13: Nash equilibrium for a game of plain complements
© 2009 Pearson Education Canada 15/42 From Figure 15.13 A game of plain complements is a game of mutual positive externalities. The more one firm produces, the higher is the price and hence the profit for the other firm. Notice the lens of missed opportunities lies above and to the right of the Nash equilibrium. This indicates the players chose strategy values that are too small in the Nash equilibrium.
© 2009 Pearson Education Canada 15/43 Mixed Strategies and Games of Discoordination Games with no Nash equilibrium are known as games of discoordination. Mixed strategies are solutions to the “no-equilibrium-in pure strategies problem.” In a mixed strategy, a player’s equilibrium consists of a probability distribution over a set of actions.
© 2009 Pearson Education Canada 15/44 Figure 15.15: Payoff matrix for a discoordination game
© 2009 Pearson Education Canada 15/45 Figure 15.16 Mixed strategy space for a discoordination game
© 2009 Pearson Education Canada 15/46 Table 15.1: Classification of outcomes and payoffs Possible Outcomes Claire’sPayoff Probability of Each Outcome Zak’s Payoff for Each Outcome (A,A)1pq0 (A,B)0 p(1- q) 1 (B,A)0 (1- p)q 1 (B,B)1 (1- p)(1- q) 0
© 2009 Pearson Education Canada 15/47 Mixed Strategies and Games of Discoordination Claire’s payoff is the probability weighted average of the payoffs associated with each outcome: Π 1 (p,q)=1(p,q) +0(p(1-q))+0((1-p)q) +1((1-p)(1-q)) Claire’s payoff is a linear function of her strategy, p: Π 1 (p,q)=(1-q)+p(2q-1) Zak’s payoff is a linear function of his strategy, q: Π 2 (p,q)= p+q(1-2p)
© 2009 Pearson Education Canada 15/48 Mixed Strategies and Games of Discoordination Claire’s best response function: 1. Her payoff increases as P increases if 2q- 1>0, or if q>1/2 and p=1 is her best response. 2. Her payoff decreases as p increases if 2q - 1<0, or if q<1/2 and p=o is her best response. 3. Her payoff doesn’t change as p increases if 2q - 1=0, or of q=1/2, and any value of p is her best response.
© 2009 Pearson Education Canada 15/49 Mixed Strategies and Games of Discoordination Zak’s best response functions: 1. q=0 is his best response if (1 - 2p) 1/2. 2. q=1 is his best response if (1 - 2p)>0, or if p 0, or if p < 1/2. 3. Any q in the interval [0,1] is best response if p = 1/2
© 2009 Pearson Education Canada 15/50 Mixed Strategies and Games of Discoordination To find the Nash equilibrium, plot the best response functions and find where they intersect. Nash equilibrium is p 0 =1/2 and q 0 = 1/2 (see Figure 15.17) Note that the equilibrium is weak in that it represents a strategy combination where no player has an incentive to deviate from his strategy given that other players do not deviate.
© 2009 Pearson Education Canada 15/51 Figure 15.17 Mixed strategy Nash equilibrium
© 2009 Pearson Education Canada 15/52 Sequential Games The extensive form is a game tree that shows which player moves when, what choices exist, and what payoffs result when the game is finished. (see Figure 15.18). Nodes are points where either decisions are made or payoffs are received. Terminal nodes are the final nodes on an extensive form game, representing the payoffs for reaching that node.
© 2009 Pearson Education Canada 15/53 Figure 15.18 The sequential word processor game
© 2009 Pearson Education Canada 15/54 Figure 15.19 Pruning branches on the word processor game
© 2009 Pearson Education Canada 15/55 Sequential Games The equilibrium path is the sequential combination of nodes in play that demonstrates the game’s equilibrium. The equilibrium is found using the backwards induction algorithm. This equilibrium is known as a subgame perfect Nash equilibrium.
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