Outlines Nash Equilibrium Simultaneous games dominant and dominated strategies Prisoner dilemma Meeting-competition clauses Sequential games Subgame perfect equilibrium
What is a game? A game consists of A set of players Rules of the games A payoff function, one for each player, telling us his payoff given actions of all players Knowledge of the players (viz., whether one firm ’ s MC is also known by the other, … ) Remark: Unless o. w. stated, we assume common knowledge among players about bullet points 1 – 3. If not, we are in a domain of asymmetric information — a forthcoming topic in our class
Nash equilibrium A Nash equil. is a complete prescription of what each player will do in every possible contingency in such a way that no player will have incentive to unilaterally deviate from his own prescription. (John Nash--Nobel Laureate 1994) Remarks: Nash equilibrium may not be unique (multiple equil.) Nash equilibrium can be mixed (penalty kick game) The game need not be symmetric. (Symmetric games are special cases.)
Output Decisions in Duopoly 2 firms producing the same product each can produce 0, 1, 2, 3 units of output at a zero marginal cost the inverse market demand function is as follows: Q. demanded 2 345 6 price161063 1.5
Output Decisions in Duopoly Payoff Table Each producing 1 unit maximizes their joint profits Each producing 2 units will be the outcome without a binding agreement
Cover story competition Each needs to determine what cover story to use for the coming issue 70 percent of potential are interested only in EOC; the other 30 percent interested only in Harbor fest; If same story, buy either one with equal chance Each cares only about its size of readership The above is commonly known between Next and Oriental Then, EOC being chosen by both is a Nash equilibrium Oriental Next EOCHarbor fest EOC35, 35 70, 30 Harbor fest 30, 70 15, 15
Dominant & Dominated Strategies A strategy is a dominant strategy for a player if that strategy gives a higher payoff than other strategies of his do, irrespective of the other player ’ s strategy. A dominant strategy may not exist A strategy is a dominated strategy for a player if that strategy gives a lower payoff than some other strategy of the player does, irrespective of the other player ’ s strategy. A dominated strategy may not exist
How to find out a Nash equilibrium? If you have a dominant strategy, use it. If you have a dominated strategy, don ’ t use it. Assume similar behavior on the part of your rivals
Cover story competition (cont.) Next and Oriental each needs to determine what cover story to use for the coming issue 70 percent of potential are interested only in EOC; the other 30 percent interested only in Harbor fest Same story, 60% among the interested readers will buy Next, the other 40% buy from Oriental Oriental NextEOCHarbor fest EOC42, 2870, 30 Harbo r fest 30, 7018, 15
Prisoners’ Dilemma Two suspects were prosecuted for robbery. Now they are under investigation separately and simultaneously. What will each of them do to minimize his/her years of imprisonment?
Prisoners’ Dilemma Neither to confess is best for both suspects when they are viewed as an entity (collective rationality). But each to confess is optimal regardless of the other does (individual rationality) Individual rationality may not lead to social optimum. Much research in oligopoly is on collusion, I.e., how firms can resolve the prisoners’ dilemma.
Application: Meeting-Competition Clauses The equilibrium outcome is (low price, low price). Collusion of (high price, high price) is unsuccessful, at least for a one-shot relationship. But collusion can be supported by meeting- competition clauses.
Meeting-Competition Clauses A meeting-competition clause states that if another firm offers a lower price, the seller will match it or release the buyer from the contract. With meeting-competition clauses, both firms choose high prices. Hence MCCs might be anti- competitive, rather than pro- competitive!!
Most-Favored-Nation Clauses A most-favored-Nation clause guarantees the buyer that the seller would not offer a lower price to any other buyer, current or future, without offering the same price decrease to the initial buyer. This rebate system creates a penalty for cheating on the cartel: If either firm deviates from the agreement by cutting its price, it would have to cut prices to all previous buyers as well!
Entry Game New Firm Enter Incumbent Accommodate Fight Price War $100,000 to New Firm $100,000 to Incumbent -$200,000 to New Firm -$100,000 to Incumbent Keep Out $0 to New Firm $300,000 to Incumbent
Entry Game Traditionally, a sequential game is represented in a matrix form like this (as if it is a simultaneous game): incumbent accommodatefight price war new firm enter100k,100k-200k,-100k keep out 0,300k Although (keep out, fight price war) is a Nash equilibrium. But we wonder if the incumbent will really implement the fighting plan if the new firm has really entered. It is not in its interest to do so…
Subgame Perfection (Selten) For sequential games, an equilibrium consists of instructions regarding players’ prescribed actions even for the point which is never reached should the equilibrium strategies are followed. Refinement/Restriction: the agents at the point is still required to act optimally even if the equilibrium does not prescribe the point to be reached. Equilibria survive such a refinement is called subgame perfection. Subgame perfect equilibria rule out incredible threats.
Poison Pills against Hostile takeovers Sharon – 1 million shares, $2 each Shareholder rights plan – allowing each shareholder to buy from the company new shares equal to his current stake at half a price. If any party acquires 100,000 or more Sharon’s shares, then management can activates the plan – but the one whose acquisition activated the rights, however, is depleted of the rights It increases the costs of acquisition, making successful hostile takeover more difficult
Sharon Hilda acquires 100,000 shares doesn’t acquire does not activates rights Hilda loses on initial stake + cost of takeover rises Poison Pills
Poison Pills (cont) Suppose the raider – Hilda – has acquired 100,000 shares and the management activates the plan. Then price of share adjusts from $2 to $v. Suppose all eligible shareholders use their rights It is dominant strategy for each shareholder to exercise his rights--the market price is v, while you can buy it at v/2 The worth of the firm = $2M + 900,000(v/2) The number of share = 1.9M Hence, each share is worth (2 + 0.45v)/1.9 =v. Hence, v = 1.38.
Poison Pills Hostile takeover is more difficult to succeed for two reasons Acquiring the whole firm is more costly -- After the plan is activated, the remaining shares is worth 1.38 x 1.8m = $2.48m, rather than 2 x 0.9m = $1.8m – the worth in the absence of poison pills The worth of the 100,000 shares is reduced – Hilda paid 2 x 100,000 = $0.2m for it. But after the plan is activated, it is worth 1.38 x 100,000 = $0.138m only.
Are Poison Pills beneficial to Shareholders? Poison Pills are takeover defenses They remove the ability of the owners (shareholders) of the firm to sell their shares to a buyer that their nominal agents – management and the board – do not like. Takeover defenses, if not strong enough to make takeovers impossible, may improve the stockholders’ bargaining position and raise the price they ultimately receive
Judicial Procedures A defendant’s fate rests in the hands of three Judges Three possible outcomes: acquittal, life in prison, and death sentence Judge A’s ranking Judge B’s ranking Judge C’s ranking BestDeath Sentence Life in Prison Acquittal MiddleLife in Prison AcquittalDeath Sentence WorstAcquittalDeath Sentence Life in Prison
Judicial Procedures (cont.) The so called status quo Judicial Procedure is used: Stage 1: determine (by majority vote) innocence or guilt by majority rule Stage 2: If innocence, acquittal automatically; if guilty, determine (by majority rule) the appropriate punishment. What will the judges do in equilibrium?
Judicial Procedures (cont.) What will the judges do under different judicial procedures? Roman Tradition: First decide if the death penalty should be imposed for this case. If not, then decide whether a life sentence is justified. If, after proceeding down the list, no sentence is imposed, then the defendant is acquitted. Mandatory Sentencing: First specify the sentence for the crime. Then determine whether the defendant should be convicted.