GAME THEORY: inside oligopoly Baye - Chapter 10

Other Non-cooperative Oligopoly Models GAME THEORY: inside oligopoly Baye - Chapter 10 • Increasing attention in business is being given to tactics and strategy to achieve competitive advantages. • We’ll view rival firm behavior as if it were a game. • Better than an analogy with war, as there need not necessarily be a loser. • Permits “thinking” about changes in strategies over time. • In oligopolistic industries, the interdependence among firms is most keenly felt. Guess who played on his HS chess team?

Business Strategy Games • When oligopolistic rivals alter their products or pricing, our firm must react or adapt. • Best would be proactive behavior that could anticipate actions. • A sequential gameis one in which there is an explicit order of play. • A sequential example is when one firm has announced a price cut, your decision to respond or not is sequential. • A simultaneous gameoccurs when all players must chose their actions at the same time.

Game TheoryWith Simultaneous Decisions • John Von Neuman & Oskar Morgenstern-- • Game Theory used to describe situations where individuals or organizations have conflicting objectives • Examples: Pricing of a few firms. Strategic Arms Race. Advertising plans for a few firms. Output decisions of an oligopoly. • Strategy– describes actions a player will take • The PAYOFF is the outcome of the strategy. • Listing of PAYOFFS appear in apayoff matrix.

Each player knows his and opponents alternatives Preferences of all players are known Single period game Sum of payoffs are zero Like a Poker Game Nash Equilibrium--none of the participants can improve their payoff Two Person, Zero Sum Game in normal-form ASSUMPTIONS PLAYER 2 c d 1, -1 3, -3 a PLAYER 1 -2, 2 0, 0 b Player 1 is the first number in each pair. We will get to {a,c} which is a Nash Equilibrium

Nash or an “Opie” Equilibriumin A Beautiful Mindpage 358 • Five brunettes and one blonde enter a bar • There are six guys watching them • If all go for the blonde, no one wins • So, is it a Nash Equilibrium if all go for the brunettes? • Ron Howard was the director and also played Opie Taylor on the Andy Griffiths Show • What is wrong with leaving the blonde alone as suggested by the movie?

Dominant Strategies • A dominant strategyis the best decision, no matter what anyone else does. It is an action (strategy) that is better in each "state of the world.” • Makes analysis easier by reducing the number of possible strategies. • Indiana Jones and the Last Crusade > what was Professor Jones’ error? • When no Nash equilibrium exists, it is useful to hide one's strategy by randomly changing strategies. • This is a mixed Nash equilibrium strategy.

Dominated Strategies • Regardless of what other players do, if a strategy always yields a lower payoff, it is a dominated strategy. Don’t take that strategy. It simplifies the game. • In the following football example, Is there a ‘Dominated Strategy’ ? If so, what is it? Payoffs in yards gained Defensive Strategies Against Run Linebackers Back Blitz Offensive Strategies Run Pass 2 6 14 8 7 10

For Player 1, strategy (a) is a dominant strategy best regardless of what others do Maximin Strategy the choice that MAXIMIZES across the set of MINIMUM possible payoffs. Best of the Worst Dominant Strategies & Maximin Strategies PLAYER 2 c d 1, -1 3, -3 a PLAYER 1 -2, 2 0, 0 b Player 1 looks for the Max { Min} as Max {1, -2} so picks Strategy-a Player 2 looks for Max { Min } as as Max {-1, -3} so picks Strategy-c

Maximin Strategy as Military Strategy Battle of the Bismarck Sea N S • Field commanders are told to make decisions based on“enemy capabilities”not on what the enemy is “likely to do” • Commanders are instructed to avoid worst possible outcomes--which is the essence of Game Theory N 2 2 US S 1 3 convoy routes rain in North Rabaul New Guinea Lae clear in South

Wilma’s payoffs appears in upper triangle and Fred’s appear in the bottom Find Maximin Solution Is it a Nash Equilibrium? Find Maximin Strategiesfor Wilma and Fred Fred c d e 5 1 -1 -5 -1 1 a b Wilma 3 7 -8 -3 -7 8 Worst for Wilma witha-strategy is -1 Worst for Wilma with b-strategy is -8 Worst for Fred with c-strategy is -5 Worst for Fred with d-strategy is -7 Worst for Fred with e-strategy is 1 best ? best

Some nations are rich in natural resources but are poor Some are poor in natural resources but are rich What causes the Swiss to be richer than the Brazilians? Some argue that political and legal institutions really do matter Ownership of property and respect for contracts Rule of law Common vs. Civil Law origins Table 4-1 page 22 What leads someone to invest? Expectations about the future? Other sources of wealth, include good health Yellow fever & Malaria hinder wealth How to lose wealth? Destroy all of the institutions that protect property and people Example: Zimbabwe The Mystery of WealthMBN Chapter 4

A Coordinated Gamenon-zero sum game PLAYER 2 • Either {a,c} or {b,d} is much better than alternatives • Two Nash Equilibria • Neither will cheat on an agreement on Volts for outlets c d 100, 100 0,0 a PLAYER 1 0, 0 100, 100 b 120-Volt and 90-Volt outlets Example: Table 10-4 page 361

Table 10-2 A Pricing Game • If a one-shot game, both have incentives to switch from {High Price, High Price} to a low price strategy • The fragility of collusion is seen again in another example of the Prisoner’s Dilemma

Noncooperative Solution both confess: {C, C} Cooperative Solution both do not confess {NC,NC} Off-diagonal represent a Double Cross Often the payoffs vary depending on the strategy choices (one period) Famous Example: The Prisoner’s Dilemma Two spies are caught & held separately Confess or Not Confess Two-Person, Non-Zero Sum Games spy 2 NC C 1 yr 15 yrs NC C 1 yr 0 yrs spy 1 0 yrs 5 yrs 15 yrs 5 yrs

Duopoly as a Prisoner’s Dilemma FIRM 2 • Even if both spies meet to agree on a cooperative solution, one may double cross. • Two firms: Decision is the amount of output [ S = small, or L = large ] • {L,L} represents normal profits S L S L 100, 100 10, 150 150, 10 20, 20 FIRM 1 MAXIMIN SOLUTION {L, L } Is it a Nash Equilibrium?

Escape From Prisoner's Dilemma: Repeated Games • If the games are repeated, there is greater expectation that firms will achieve the cooperative solution. • Each firm "shows" by its behavior each period that it wants to cooperate. • Firms that expand production "show" that they do not want to cooperate. • “Tit for Tat” strategy, is a matching strategy.

Duopoly as a Multiperiod Game • The single period(One-Shot) games predicts that there will be competition • But duopolists are likely to have many periods in which to compete • Multiple periodsallow for “Punishment” or retribution not found in single period games. • We would expect that collusion is More Likely to Succeed, the greater chance for more periods

Unstable Games:No Nash Equilibrium Is Found Fred c d • In the Wilma-Fred Game here, Maximin Strategies lead to solution {b, c} • But Wilma has an incentive to switch to strategy-a • Then Fred has an incentive to switch to strategy-d, etc., etc. a b 3, - 3 1, - 1 2, - 2 4, - 4 Wilma There is no, single stable equilibrium Each player may elect a random strategy

Games without a Nash Equilibrium have no pure strategy They tend to be best solved by randomizing one’s actions Mixed randomized strategy Examples Timing of sales or introduction of ad campaigns Hitters and Pitchers 40% 20% 20% 40% Pitcher Fast Curve Mixed Strategies Fast Curve If a hitter thinks that the pitcher will throw fastballs 50:50, then will hit only 300. But if he assumes that fastballs are more likely, the hitter can do better.

Suppose Pitcher throws fastballs 70% of the time. • If the hitter anticipates fastballs100% of the time, and the pitcher continues to throw fastballs 70% of the time, what will be the hitter’s batting average? • (.70)(.40) + (.30)(.20) = .28 + .06 = .34 or 340.

Unpredictability? Or Mixed Strategies • If the IRS audits Schedule A “gifts in kind” only when gifts are greater than $1,000, then what? • Suppose a batter is known for always “looking at” the first pitch, then what? • Can be an “optimal amount” of randomness (tennis serves fore- & back-hand, for example) • Most ‘games’ lend themselves to known preferences and actions, but sometimes a reputation for fast change can be useful.

Trade restrictions are proposed to ‘protect jobs’ and help workers Restrictions in steel imports to help steelworkers maintain high pay They also have spillover effects in other industries. Fewer imports would tend to lead to fewer exports, since imports tend ultimately to paid for in exports. Restrictions on imports then hurts exporters generally. Helping some to hurt others is not a Pareto Optimal The $750,000 JobChapter 31 of MBN

Voluntary trade restrictions of Japanese automakers raised price $1,500 for their cars and US cars by $600 Cost was $6.6 Billion helping 26,000 workers at a cost of $250,000 per job Tariffs on apparel saved 116,000 jobs at a cost of $45,000 each Tariffs on CB radios saved 600 workers at a cost of $85,000 each Tariffs on steel cost an astonishing $750,000 per job saved But the cost is greater when we know that import restrictions lose jobs in exporting industries Autos, Apparel, CB Radios & the Steel Industry

Infinitely Repeated Games • Successful businesses last for a long time • A decision in one period therefore can have long lasting consequences • Simplified version of model in Baye, pages 367-8. • If in a duopoly, you coordinate and stay faithful, you earn p each period. The present value of a constant stream of these payments is p/i, where i is the interest rate. • If cheat, get gain G, but thereafter the cooperation is broken and only get 0. • We do not break the agreement if G < p/i • We cooperate whenever the one-time gain from breaking the agreement is less than the present value of the loss in future: pCheat– pCoop < (pCoop – pN)/i The one-shot Nash payout

Analysis & Trigger Strategies • If a game lasts a long time, the punishment is larger, and more likely to collude. • If the game lasts only one period, it is best to cheat immediately. • If a finitely repeated game game lasts two periods, each has an incentive to be the first to cheat, and collusion tends to fall apart. • Grim Trigger Strategy – punish forever • The unfaithful spouse ends in divorce • Trembling Hand Trigger Strategy – I may or may not punish • The unfaithful spouse might get divorced

If all firms picked price P When just a few firms, profits are enhanced if all reduce output to q But each firm has incentives to “cheat” by selling more Cooperative Oligopoly Models MC MC P D MR incentive to cut price q QM Typical Firm Industry

Factors Affecting Collusion in Pricing Games 1.Number and Size Distribution of Sellers. Collusion is more successful with few firms or if there exists a dominant firm. 2. Product Heterogeneity. Collusion is more successful with products that are standardized or homogeneous 3. Cost Structures. Collusion is more successful when the costs are similar for all of the firms in the oligopoly. 4. Size and Frequency of Orders. Collusion is more successful with small, frequent orders. 5. Secrecy and Retaliation. Collusion is more successful when it is difficult to give secret price concessions. 6. Social Structure and history of the Industry. Collusion is more successful when industry executives often meet together

Applications of Finitely-Lived Games • Resignation and quitting behavior • If working, shirking has a large potential damage • If you have announced you are leaving in a few weeks, the loss due to firing is less • Snake-oil salesman • If you have a store, your game is infinitely-lived • If come to town to sell a secret elixir, but you move on, Caveat Emptor.

Multistage Games & Game TreesAn Illustration of a Sequential Game • A game tree is like a decision tree. It is a schematic diagram of decision nodes. • Solutions to games parallels board games like chess. • One way to solve a decision problem is to use end-game reasoning, where we start with the final decision and use backward induction to find the best starting decision on the game tree.

Figure 10 – 1 page 378Players A & B {10, 15} {5, 5} {0, 0} {6, 20} • Decision nodes • Like moves of a player or a price offer • Payoffs with A given first • A will decide to pick the subgame of UP • Look at end-game payoffs • Backward induction reasoning • Only credible threat can work B up A down B “If you’re not home by midnight, I’ll burn the house down.” isn’t credible.

GAME THEORY: inside oligopoly Baye - Chapter 10