Game Theoretic Analysis of Oligopoly.

lr L R 0000 L R 1 22 The Lane Selection Game Rational Play is indicated by the black arrows

lr L R 0000 L R 1 22 l, r and L, R are called choices Decision Nodes Terminal Nodes Top number in matrix is player 1’s payoff

A Nash Equilibrium is a strategy pair such that each player plays his/her best strategy, given the strategy of the other player A best strategy of a player yields the maximum payoff to him/her, for a specific strategy chosen by the other player A strategy is a contingency plan that specifies a choice at each possible decision node

A Dominant Strategy is such that it is the best strategy of the player no matter what strategy is chosen by the other A mixed strategy is a combination of two or more strategies, each chosen with some positive probability A pure strategy is the choice of a strategy with probability 1

3. {R if l}), ({L if r}, 4.{L if l},and {R if r} Nash Equilibria [l, {L if l}, {R if r}] and [r, {L if l} {R if r}] Player 2’ s strategies 1.({L if l}, {L if r}), 2.{R if l},and {R if r} Player 1’ s strategies are l and r

1 ht H T H T The ‘Matching Pennies’ Game

The unique Nash Equilibrium (in mixed strategies)is that each player mixes up each pure strategy with 0.5 probability Player 2 has two pure strategies: Hand T Player 1 has two pure strategies: h and t

y n Y N 0000 Y N The unique dominant strategy Nash Equilibrium is (y,Y) A game of imperfect Information The Prisoners’ Dilemma Y y stand for compete N n stand for collude

y n Y N 0000 Y N The Prisoners’ Dilemma A game of Perfect Information The only play is (y, Y)

T M B L RC L R C LRC

1 2 Enter Stay Out Tough Soft 3m -1m 2m 0 7m 1- Entrant 2- Incumbent 1: Stay Out 2: Tough if Enter 1: Enter 2: Soft if Enter The two Nash Equilibria are Credible Threat Equilibrium

Finitely Repeated Games Prisoners’ Dilemma

y n Y N 0000 Y N The Prisoners’ Dilemma A game of Perfect Information Player 1 plays y and player 2 plays Y if y and Y if n at the only Nash Equilibrium Y y stand for compete N n stand for collude Game 2

y n Y N 0000 Y N The Prisoners’ Dilemma A game of Perfect Information Y y stand for compete N n stand for collude Game 200 Player 1 plays y and player 2 plays Y if y and Y if n at the only Nash Equilibrium

Finite Sequence of Entry Games

1 2 Enter Stay Out Tough Soft 3m -1m 2m 0 7m 1- Entrant 2- Incumbent 1: Stay Out 2: Tough if Enter 1: Enter 2: Soft if Enter The two Nash Equilibria are Game with two sequential entries

1 2 Enter Stay Out Tough Soft 3m -1m 2m 0 7m 1- Entrant 2- Incumbent 1: Stay Out 2: Tough if Enter 1: Enter 2: Soft if Enter The two Nash Equilibria are Game with two hundred sequential entries