Game Theory Exam 1 Review PDF

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Shaista Dhanesar GAME THEORY EXAM 1 REVIEW Game Theory - The study of strategic situations or (“games”) - A normal form game consists of: o Players: who are the decision makers  A finite set of players N = {1, 2, …, n), indexed by i ( i = 1, 2, . . . ,n) o Strategies: what decisions/ actions are available to each player  For each player, i, a set of strategies Si, where each element of Si is a strategy for i o Payoffs: how does each payer evaluate any possible combination of decisions  πi (S1, . . . Sn) Monopoly - Suppose a monopolist sells a good or service - Inverse demand p = a – bq where a>0 and b>0 - Costs c(q) = cq - Profits π, where π= pq – cq = (p-c)(q) - Can be written as a normal form game: o Players: N = {1} o Strategies: Si = {q| q > 0}  Can choose either q or p but not both; other dimension is determined by consumers o Payoffs: πi = (p-c)(q) = ((a-bq)-c)(q) = (a-bq-c)(q) - Best response of a monopolist is simply the profit maximizing output - Qm = (a-c)/2b Pm = (a+c)/ 2 MR = a-2bQ (p. 14) Oligopoly - A collection of firms (2 or more) in a particular industry - Firms in oligopoly must account for the behavior of competing firms Cournot (quantity) Competition - 2 firms produce a homogeneous good - Ex. p. 17 - How to solve a Cournot problem (p. 38) o Write down profit functions o Take derivatives with respect to what each firm controls (q) o Solve for variables  NE o Use NE to solve for π Bertrand (price) Competition - 2 firms produce a homogeneous good - Demand shared equally among lowest priced firms - Bertrand model (p. 19) - Bertrand Paradox: Can firms mitigate “cutthroat” competitive pressures in price competition?

Shaista Dhanesar o Possible solutions:  Product differentiation  Capacity constraints  Dynamic competition  Low price guarantee Low Price Guarantee (p. 34) - Firms tend to be retailers - Common wholesale price, so no cost advantages - Identical products so loss of price sensitivity - Capacity constraints probably not a big issue - Low price guarantees seem to raise competitive pressures even more - Ex. P. 34 Product Differentiation - If the cost of producing cosmetically different goods is low, manufacturers can increase hassle costs - With many different variants of a product, it is hard for customers to be sure products are “identical” - It is also harder for retailers to specify the product whose price they’re guaranteeing - Key advantage of differentiate = uniqueness  market power Dominance (p. 8) - A strategy is dominant for a player if it gives a strictly higher payoff than every other strategy for the plater, for any strategy of the other player - A rational player will always play the dominant strategy - Ex. Prisoner’s dilemma (p.7), Ex. (p. 10) - Ex. Market share game (p. 11) o For both column and row player, D is dominant strategy o (D,D) is the only rational outcome in this game) - If Si strictly dominates every other strategy Si’, then it is strictly dominant - If Si weakly dominates every other strategy Si’, then it is weakly dominant Best Response (p. 9) - A strategy Si by player 1 is a best response to a list of strategies S-I played by i’s opponent if πi (Si, S-i) > πi (S’I, S-i) Nash Equilibrium (p. 9) - A choice (actions, probabilities) profile such that nobody has an incentive to deviate; every player is best responding to the choices of others - If an outcome is predicted based on the assumption that players are rational, no individual player should be able to improve their payoff by behaving differently, given how others are predicted to behave - You cannot unilaterally do better by switching - An equilibrium is a strategy profile: a strategy for each and every player (p. 12) - Nash may not be reached immediately, but eventually players will converge there - No regrets - Does not need a contract to be enforced - Ex. Battle of the sexes/ coordination game (p. 13)

Shaista Dhanesar - Ex. Chicken game/ anti-coordination game (p. 14) Mixed Strategies - Ex. Battle of the Sexes (p. 21) o FF and OO are best responses, but could also flip a coin o 3 strategies: F, O, coin toss o After the coin has been tossed, a player will still be going t F or O  Randomizing does not enlarge set of eventual actions  It does enlarge the set of initial choices (clearly distinct from F or O) o O and F are pure strategies o Coin toss strategy is a mixed strategy Mixed Strategies – Expected Payoff - Weight the payoff to each pure strategy profile by the probability with which that profile is played, then add up the weighted payoffs - Ex. P. 22 - Mixed Strategy formula (p. 40) o Allocate probabilities to each strategy of P1 o Write E[π] for each strategy of other player o Make player indifferent between his strategy by choosing probabilities o Solve probabilities o Repeat for plater 2 The Indifference Principle - In a mixed action equilibrium, players are indifferent among the actions they are mixing on (p. 33) - Consider a mixed strategy Si by plater i - The support of Si consists of all pure strategies that are played with positive probability - Si is a best response to S-I iff each pure strategy in its support is itself a best response to S-i - Never play strictly dominant strategies with positive probabilities - Ex. P. 27 - Without mixed strategies, nash equilibria need not exist - Consider game of matching pennies (p. 29) o Appears to be no pure strategy nash equilibrium - If you need to be unpredictable: o Don’t play recognizable patterns o If the cost of some activity goes up, remember that you are dealing with strategic competitors  A small change in the level of use of that activity may have dramatic consequences on your payoffs  Ex. P. 33...