Other Issues in Game Theory

1. Other Issues in Game Theory Business Negotiations Contracts

2. Dominant Strategy • Regardless of whether FIRM 2 chooses strategy A, B, or C, FIRM 1 is better off choosing “a”! • “a” is Player 1’s Dominant Strategy! • Dominance is a solution strategy --- but doesn’t always lead to “resting point” --- so we introduce the Nash solution --- Nash equilibrium

3. Putting Yourself in your Rival’s Shoes • What should FIRM 2 do? • 2 has no dominant strategy! • But 2 should reason that 1 will play “a”. • Therefore 2 should choose “C”.

4. The Outcome • This outcome is called a Nash equilibrium: • “a” is player 1’s best response to “C”. • “C” is player 2’s best response to “a”.

5. A NASH EQUILIBRIUM IN WHICH EVERY PLAYER PLAYS A PURE STRATEGY IS CALLED A PURE STRATEGY NASH EQUILIBRIUM ANY STRATEGY THAT IS NOT COMPLETELY DETERMINISTIC, BUT INSTEAD INVOLVES CHANCE (RANDOMIZATION), IS CALLED A MIXED STRATEGY --- SO A NASH EQUILIBRIUM IN WHICH AT LEAST ONE PLAYER PLAYS A MIXED STRATEGY IS CALLED A MIXED STRATEGY NASH EQUILIBRIUM COORDINATION GAMES USUALLY RESULT IN MIXED STRATEGIES AS WELL AS PURE NASH

6. A MORE SIMPLIFIED COORDINATION GAME

7. There are 2 pure Nash {both firms choose 210 v} and { both firms choose 120 v } There is 1 mixed strategy {1/2[210v], ½[120v]} for each firm

8. Two stock brokers, Funk and alias, Naylor, are indicted by the N.Y. Attorney General for allegedly making use of illegal inside information --- but the evidence is weak The attorney general brings them in to interrogate, one at a time Both Funk and Naylor have two possible strategies, confess or remain mum 4 possible strategies are outlined in the normal form game below Naylor confess mum confess Funk mum

9. SO, WHAT’S GOING TO HAPPEN HERE? THE DOMINANT STRATEGY FOR BOTH BROKERS IS TO CONFESS! BUT EACH IS DOING WORSE THAN IF THEY COULD TRUST EACH OTHER AND ONLY GET 4 YEARS BASED ON THE WEAK EVIDENCE BY REMAINING MUM (OH,OH, HERE COMES THE ROLE OF THE DEFENSE ATTORNEYS!) THIS IS THE PRISONER’S DILEMMA GAME Naylor confess mum confess Funk mum

10. Now recall the problems with the “Sweezy” or Kinked demand curve oligopoly case we introduced in Ch. 9 If one firm reduces price, the rival firm will match this action by also reducing price, but will not match price increases Suppose now we have two firms, Simmons and israelsen who react on pricing of their product and reap the profits (in $millions) given in the normal form given below Israelsen Price = $1,000 Price = $2,000 Price = $2,000 Simmons Price = $1,000

11. What is going to happen here? If Simmons and Israelsen form some sort of cartel --- then there is incentive to not follow through on the agreement, however supposed to be binding The Nash is that both lower their price from $2,000 to $1,000 in hopes of capturing the market Israelsen Price = $1,000 Price = $2,000 Price = $2,000 Simmons Price = $1,000

12. Now suppose they offer a “most favored customer clause” , whereby a customer who buys early at a high price gets a rebate if price is later set at a lower price --- the rebate will lower profits ---- so the payoff is now given below Israelsen Price = $1,000 Price = $2,000 Price = $2,000 Simmons Price = $1,000

13. We now get two pure Nash {both firms set price at $2,000} and {both firms set price at $1,000} And then we get a mixed strategy { both firms choose price = $2,000 with probability 4/5, and choose price = $1,000 with probability 1/5} So the weight on the choice suggest the most favored customer clause provides incentives to hold at the $2,000 price --- a possible way out of the kinked demand curve Israelsen Price = $1,000 Price = $2,000 Price = $2,000 Simmons Price = $1,000

14. Suppose Dell and GE are considering engaging in a joint venture. Each will have to invest $12 million in assets that are of no value outside the project (specialized or specific investments and costs) If both firms act in accord with their promises, the annual “economic profit” to each firm is $3 million If one or both do not act in accord with promises, then the annual profit is as shown in the following normal form Dell Accord No accord Economic profit in $millions Accord GE No accord BOTH FIRMS HAVE THE OPTION TO NOT PLAY THE GAME

15. Will a contract be drawn up and signed by both parties? Yes --- the Nash equilibrium is (3,3) Without such a contract, each firm would have incentive to go their separate ways after investment ---- managers may be reluctant to take the risk of investment --- so trust is what binds the contract --- coordination Economic profit is profit above what could have been earned in alternative investment opportunities for the $12 million Dell Accord No accord Accord GE No accord THIS IS NOT A PRISONER’S DILEMMA GAME

16. WHEN IS A THREAT CREDIBLE? Firms often signal to each other to indicate intentions -- Youngberg announces it is moving to lower price --- McKenna then intends to significantly lower its own price signaling willingness to engage in price war {see the payoff matrix below} But McKenna’s threat is not credible --- profits at high price are more than profits at low price --- dominant strategy for McKenna is high price, irrespective of Youngberg’s price ---- Nash is (7,11) Youngberg Low price High price Low price McKenna High price