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Chapter 15 Introduction to Game Theory. 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.
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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.
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.
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.
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.
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.
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.
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.
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.
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.
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?
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).
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: • There are many real life applications. • 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).
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.
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.
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.
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.
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.
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.
Games of Plain Substitutes with Simultaneous Moves • The cross-effects in the payoff functions are negative. • There exists mutual negative externalities. • y10 and y20 are the Nash equilibrium values of the strategies. • From the Nash equilibrium, y10 is a best response to y20
Games of Plain Substitutes with Simultaneous Moves (continued) Y10 solves the constrained maximization problem: Maximize by choice of y1and y2 п1 (y1, y2) < y2 = y20 Indifference curve п1 (y1, y2) is tangent to the constraint at the Nash equilibrium (y10, y20) in Figure 15.14. Because п1 (y1, y2) decreases as y2 increases, this indifference curve must lie below the line y2 = y20 elsewhere.
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 y1 = y1 at (y10, y20). Elsewhere it must lie to the left of the line y1 = y10 and the set of strategy combinations. • Two’s preferences to the Nash equilibrium lie to the left of this indifference curve.
Figure 15.14 Nash equilibrium for a game of plain substitutes
From Figure 15.14 • 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.
Mixed Strategies and Games of Discoordination • Claire’s payoff is the probability weighted average of the payoffs associated with each outcome: Π1(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: Π1(p,q)=(1-q)+p(2q-1) • Zak’s payoff is a linear function of his strategy, q: Π2 (p,q)= p+q(1-2p)
Mixed Strategies and Games of Discoordination • Claire’s best response function: • Her payoff increases as P increases if 2q-1>0, or if q>1/2 and p=1 is her best response. • Her payoff decreases as p increases if 2q - 1<0, or if q<1/2 and p=o is her best response. • 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.
Mixed Strategies and Games of Discoordination • Zak’s best response functions: • q=0 is his best response if (1 - 2p)<0, or if p > 1/2. • q=1 is his best response if (1 - 2p)>0, or if p < 1/2. • Any q in the interval [0,1] is best response if p = 1/2
Mixed Strategies and Games of Discoordination • To find the Nash equilibrium, plot the best response functions and find where they intersect. • Nash equilibrium is p0 =1/2 and q0 = 1/2 (see Figure 15.21).
