© 2005 Pearson Education Canada Inc Chapter 15 Introduction to Game Theory

© 2005 Pearson Education Canada Inc 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.

© 2005 Pearson Education Canada Inc Game Theory: Basic Definitions  Players-entities like individuals/firms that make choices.  Strategies-the choices made by the players (output/pricing, etc.).  Strategy combinations-a list of strategies for each player.  Payoff-the outcome (utility, profit, etc.) from selecting a strategy.

© 2005 Pearson Education Canada Inc Game Theory: Basic Definitions  Best response function-the player’s best response given the strategies of other players.  Equilibrium strategy combination-a strategy combination where every player’s strategy is the best response to the strategy of all other players.

© 2005 Pearson Education Canada Inc 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.

© 2005 Pearson Education Canada Inc Game Theory: Basic Definitions  Normal forms-simply represents the outcomes in payoff matrix (connects the outcomes in an obvious way).  Extensive form description-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.

© 2005 Pearson Education Canada Inc Game Theory: An example  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 player, regardless of the actions of other players, it is a dominated strategy.

© 2005 Pearson Education Canada Inc Figure 15.1 A movement game

© 2005 Pearson Education Canada Inc From Figure 15.1  For player 2, the strategy Middle is dominated by the strategy Right.  When you find a dominated strategy, it can be eliminated from the game.  Therefore, Figure 15.1 becomes Figure 15.2.

© 2005 Pearson Education Canada Inc Figure 15.2 Game with dominated strategy award

© 2005 Pearson Education Canada Inc From Figure 15.2  For player 1, the Up strategy dominates both Middle and Down.  For player 1, Up is therefore a dominant strategy.  The Middle and Down rows can be eliminated from player 1’s strategy.  This leaves the game shown in Figure 15.3.

© 2005 Pearson Education Canada Inc Figure 15.3 Game with last dominated strategy

© 2005 Pearson Education Canada Inc From Figure 15.3  Player 1 has no choice but to move Up.  For Player 2, the dominant strategy is to move Left.  (Up, Left) or 4,3 * is therefore the equilibrium payoff.  It is a Nash equilibrium where both players will settle on a strategy and not want to move.

© 2005 Pearson Education Canada Inc 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?

© 2005 Pearson Education Canada Inc Figure 15.4 The prisoner’s dilemma

© 2005 Pearson Education Canada Inc 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).

© 2005 Pearson Education Canada Inc Figure 15.5 Nash equilibrium in the PD game

© 2005 Pearson Education Canada Inc From Figure 15.5  The arrows meet where both Petra and Ryan fink (Fink, Fink) and this is the equilibrium for the game.  Interesting aspects of the prisoner’s dilemma: 1. There are many real life applications. 2. The equilibrium results form a dominant strategy for both players. 3. The equilibrium outcome is not Pareto- Optimal (both would be better off if they both remained silent).

© 2005 Pearson Education Canada Inc Coordination Games  Often situations may have no equilibrium or they may have multiple equilibria.  In these situations, other forms of behaviour must arise for a solution to be found.

© 2005 Pearson Education Canada Inc 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.

© 2005 Pearson Education Canada Inc Figure 15.8 Choosing a word processor

© 2005 Pearson Education Canada Inc 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.

© 2005 Pearson Education Canada Inc 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.

© 2005 Pearson Education Canada Inc 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.

© 2005 Pearson Education Canada Inc 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 compliments, the players impose positive externalities on each other.

© 2005 Pearson Education Canada Inc 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

© 2005 Pearson Education Canada Inc 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 ) < y 2 = y 2 0 Indifference curve п 1 (y 1, y 2 ) is tangent to the constraint at the Nash equilibrium (y 1 0, y 2 0 ) in Figure Because п 1 (y 1, y 2 ) decreases as y 2 increases, this indifference curve must lie below the line y 2 = y 2 0 elsewhere.

© 2005 Pearson Education Canada Inc 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 the figure.  For Two’s indifference curve through the Nash equilibrium. It must be tangent to the line y 1 = y 1 at (y 1 0, y 2 0 ). Elsewhere it must lie to the left of the line y 1 = y 1 0 and the set of strategy combinations.  Two’s preferences to the Nash equilibrium lie to the left of this indifference curve.

© 2005 Pearson Education Canada Inc Figure Nash equilibrium for a game of plain substitutes

© 2005 Pearson Education Canada Inc From Figure  All strategy combinations in the Lense of Missed Opportunity are preferred by both players to the Nash equilibrium.  When players impose mutual negative externalities on one another, they produce too much and would be better off cutting back on their strategy values.

© 2005 Pearson Education Canada Inc Mixed Strategies and Games of Discoordination 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

© 2005 Pearson Education Canada Inc 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)

© 2005 Pearson Education Canada Inc 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.

© 2005 Pearson Education Canada Inc 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

© 2005 Pearson Education Canada Inc 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 15.21).

© 2005 Pearson Education Canada Inc Figure Mixed strategy Nash equilibrium