Game Theory Introduction Behavior of competitors or impact

Game Theory

Introduction • Behavior of competitors, or impact of own actions, cannot be ignored in oligopoly • Managers maximize profit or market share by outguessing competitors • Insight into oligopolistic markets by using GAME THEORY (Von Neumann and Morgenstern in 1950): designed to evaluate situations with conflicting objectives or bargaining processes between at least two parties.

Types of Games • Normal Form • Simultaneous (act without knowing other player’s strategy) • One Shot • Zero Sum (market share) • Single Stage vs. Extensive Form vs. Sequential (one player moves after observing others) vs. Repeated (infinite and finite with uncertain and certain final period) vs. Non-zero Sum (profit maximization) vs. Multistage

A Normal Form Game Elements of the game: Player 1 Planed decision or actions 12, 11 11, 10 10, 15 Players Strategies or feasible actions Payoff matrix Player 2 11, 12 10, 11 10, 13 14, 13 12, 12 13, 14 Results from strategy dependent on the strategies of all the players

Dominant Strategy • Regardless of whether Player 2 chooses A, B, or C, Player 1 is better off choosing “a”! (Indiana Jones and the Holy Grail) • “a” is Player 1’s Dominant Strategy! Player 2 Player 1 12, 11 11, 10 10, 15 1’s best strategy a 11, 12 10, 11 10, 13 a 14, 13 12, 12 13, 14 a 2’s best strategy c c a

The Outcome • What should player 2 do? • 2 has no dominant strategy, but should reason that 1 will play “a”. • Therefore 2 should choose “C”. Player 1 Player 2 12, 11 11, 12 11, 10 10, 11 14, 13* 12, 12 10, 15 10, 13 13, 14 • This outcome is called a Nash equilibrium (set of strategies were no player can improve payoffs by unilaterally changing own strategy given other player’s strategy) • “a” 1’s best response to “C” and “C” is 2’s best response to “a”.

Best Response Strategy • • • Try to predict the likely action of competitor to identify your best response: Conjecture choice of rival Select your own best response Was conjecture reasonable or Look for dominant strategies Put yourself in your rival’s shoes

Market-Share Game Equilibrium • Two managers want to maximize market share (zero-sum game) • Strategies are pricing decisions • Simultaneous moves • One-shot game Manager 1 Manager 2 Nash Equilibrium

Dominated Strategy • Dominance exception rather than rule • In absence of dominance it might be possible to simplify the game by eliminating dominated strategy (never played: lowest payoff regardless of other player’s strategy) • Steelers trial by 2, have the ball & enough time for 2 plays • Payoff matrix in yards gained by offense: no dominant strategy • Pass dominant offense without Blitz (dominated defense) Offense Defense Best Offense 2 6 14 Best Defense Run 8 7* 10 Pass Run

Maximin or Secure Strategy In absence of dominant strategy risk averse players may abandon Nash or best response (*) and seek maximin option (^) that maximizes the minimum possible payoff. This is not design to maximize payoff but rather to avoid highly unfavorable outcomes (choose the best of all worst). Firm 1 Firm 2 Best for 1 Min for 2 New 6 None 3 Best for 2 New 6 None 3 Min for 1 New 3 None 2 None 3 New 2 Board of Getty Oil agreed to sell 40% stake to Pennzoil @ $128. 5 in Jan 1984. Board of Getty Oil subsequently accepted Texaco’s offer for 100% @ $128. Pennzoil sued Texaco for breach of contract & received $10 bill jury award in 1985. Texaco appealed. Before Supreme Court’s decision, they settled for $3 bill in 1987.

Examples of Coordination Games • Industry standards • size of floppy disks • size of CDs • etc. • National standards • electric current • traffic laws • etc.

A Coordination Problem: Three Nash Equilibria! Player 1 Player 2

Key Insights: • In some cases one-shot, non-cooperative games result in undesirable outcome for individuals (prisoner’s dilemma) and some times for society (advertisement). • Communication can help solve coordination problems. • Sequential moves can help solve coordination problems. • Time in jail, Nash (*) and Maximin (^) equilibrium in Prisoner’s dilemma. Suspect 1 Suspect 2 Best for 1 Max for 2 Confess Do Not Confess Best for 2 Confess Do Not Max for 1 Confess

One-Shot Advertising Game Equilibrium • Kellogg’s & General Mills want to maximize profits • Strategies consist of advertising campaigns • Simultaneous moves • One-shot interaction • Repeated interaction Kellogg’s General Mills Nash Equilibrium

Repeating the game 2 times will not improve outcome • In the last period the game is a one-shot game, so equilibrium entails High Advertising. • Period 1 is “really” the last period, since everyone knows what will happen in period 2. • Equilibrium entails High Advertising by each firm in both periods. Kellogg’s • The same holds true if we repeat the game any known, finite number of times. General Mills * Nash Equilibrium

Can collusion work if firms play the game each year, forever? • Consider the “trigger strategy” by each firm: • “Don’t advertise, provided the rival has not advertised in the past. If the rival ever advertises, “punish” it by engaging in a high level of advertising forever after. ” • Each firm agrees to “cooperate” so long as the rival hasn’t “cheated”, which triggers punishment in all future periods. • “Tit-for-tat strategy” of copying opponents move from the previous period dominates “trigger strategy” for: • Simple to understand • Never invites nor rewards cheating • Forgiving: allows cheater to restore cooperation by reversing actions

Suppose General Mills adopts this trigger strategy. Kellogg’s profits? Cooperate = 12 +12/(1+i) + 12/(1+i)2 + 12/(1+i)3 + … Value of a perpetuity of $12 paid = 12 + 12/i at the end of every year Cheat = 20 +2/(1+i) + 2/(1+i)2 + 2/(1+i)3 + … = 20 + 2/i Kellogg’s General Mills

Kellogg’s Gain to Cheating: • Cheat - Cooperate = 20 + 2/i - (12 + 12/i) = 8 - 10/i • Suppose i =. 05 • Cheat - Cooperate = 8 - 10/. 05 = 8 - 200 = -192 • It doesn’t pay to deviate. • Collusion is a Nash equilibrium in the infinitely repeated game! Kellogg’s General Mills

Benefits & Costs of Cheating • Cheat - Cooperate = 8 - 10/i • 8 = Immediate Benefit (20 - 12 today) • 10/i = PV of Future Cost (12 - 2 forever after) • If Immediate Benefit > PV of Future Cost • Pays to “cheat”. • If Immediate Benefit PV of Future Cost • Doesn’t pay to “cheat”. Kellogg’s General Mills

Key Insight • Collusion can be sustained as a Nash equilibrium when game lasts infinitely many periods or finitely many periods with uncertain “end”. • Doing so requires: • • Ability to monitor actions of rivals Ability (and reputation for) punishing defectors Low interest rate High probability of future interaction

Real World Examples of Collusion: Garbage Collection Industry Homogeneous products Bertrand oligopoly Known identity of customers Known identity of competitors Firm 2 One-Shot Bertrand (Nash) Equilibrium Firm 1 Firm 2 Repeated Game Equilibrium Firm 1

Real World Examples of Collusion: OPEC One-Shot Cournot (Nash) Equilibrium Repeated Game Equilibrium Assuming a Low Interest Rate Saudi Arabia • Cartel founded in 1960 by Iran, Iraq, Kuwait, Saudis and Venezuela: “to co-ordinate and unify petroleum policies among Members in order to secure fair and stable prices” • Absent collusion: PCompetition < PCournot (OPEC) < PMonopoly Venezuela

OPEC’s Demise Low Interest Rates High Interest Rates

Simultaneous-Move Bargaining • • Management and a union are negotiating a wage increase. Strategies are wage offers & wage demands. Simultaneous, one-shot move at making a deal. Successful negotiations lead to $600 million in surplus (to be split among the parties), failure results in a $100 million loss to the firm and a $3 million loss to the union. • Experiments suggests that, in the absence of any “history, ” real players typically coordinate on the “fair outcome” • When there is a “bargaining history, ” other outcomes may prevail Three Nash Equilibria in Normal Form Management Union

Single Offer Bargaining • Now suppose the game is sequential in nature, and management gets to make the union a “take-it-orleave-it” offer. • Analysis Tool: Write the game in extensive form • • • Summarize the players Their potential actions Their information at each decision point The sequence of moves and Each player’s payoff

To get The Game in Extensive Form Step 1: Management’s Move Step 2: Add the Union’s Move Step 3: Add the Payoffs Accept Union Reject 100, 500 -100, -3 10 Firm 5 Union Accept Reject 1 Union Accept Reject 300, 300 -100, -3 500, 100 -100, -3

Step 4: Identify Nash Equilibriums Outcomes such that neither player has an incentive to change its strategy, given the strategy of the other Accept Union 100, 500 Reject -100, -3 Accept 300, 300 10 Firm 5 Union Reject 1 Accept Union Reject -100, -3 500, 100 -100, -3

Step 5: Find the Subgame Perfect Nash Equilibriums Outcomes where no player has an incentive to change its strategy, given the strategy of the rival, that are based on “credible actions”: not the result of “empty threats” (not in its “best self interest”). Union Accept 100, 500 Reject -100, -3 10 Firm 5 Union Accept Reject 1 Union Accept Reject 300, 300 -100, -3 500, 100 -100, -3

Re-Cap • In take-it-or-leave-it bargaining, there is a first-mover advantage. • Management can gain by making a take-it or leave-it offer to the union. But. . . • Management should be careful, however; real world evidence suggests that people sometimes reject offers on the basis of “principle” instead of cash considerations.

Pricing to Prevent Entry: An Application of Game Theory Two firms: an incumbent and potential entrant. Identify Nash and then Subgame Perfect Equilibria. Hard -1, 1 Incumbent Enter Soft Entrant Out 5, 5* 0, 10 Establishing a reputation for being unkind to entrants can enhance long-term profits. It is costly to do so in the short-term, so much so that it isn’t optimal to do so in a one-shot game.

The Value of a Bad Reputation: Price Retaliation • In early 1970 s General Foods’ Maxwell House dominated the non-instant coffee market in the Eastern USA, while Proctor & Gamble’s Folgers dominate Western USA. • In 1971 P&G started advertising & distributing Folgers in Cleveland Pittsburgh. • GF’s immediately increased advertisement & lowered prices (sometimes below cost) in these regions & midwestern cities (Kansas City) where both marketed. • GF’s profit dropped from 30% in 1970 to – 30% in 1974. When P&G reduced its promotional activities, GF’s price increased and profits were restored.