School of Economics, Mathematics and Statistics INDUSTRIAL ECONOMICS, WITH APPLICATIONS TO E-COMMERCE An Option for MSc Economics and MSc E-Commerce Autumn Term 2003 1. Strategic Interaction and Oligopoly Aims and Objectives: This section of the course aims to study the standard models of oligopoly. This will involve Game theoretic tools Oligopoly theory: Bertrand and Cournot competition Repeated games and collusion Imperfect monitoring Readings and Reference Jean Tirole, The Theory of Industrial Organization, MIT, 1998. Chapter 5 (Sections 1, 2, 3.2.1, 4, 5), Chapter 6 (Sections 1, 3.1, 3.3) *Kreps, D & J Scheinkman (1983) 'Quantity Precommitment and Bertrand Competition yield Cournot outcomes', Bell Journal of Economics, 14, 326-37 Outline 1 Game theoretic tools 2 Some standard models of oligopoly Bertrand equilibrium: the nature of price competition: the paradox, and possible resolutions Cournot equilibrium: two firms; reaction curves: many firms, and limiting results Outline of Kreps-Scheinkman model: price competition with capacity constraints 3 Collusion: explicit collusion and tacit collusion in repeated games. Factors inhibiting and encouraging collusion 4 Outline of the Green and Porter model
Game-theoretic tools October 5, 2003 Whatyouwilleventuallyneedtoknow 1. Games in extensive and strategic forms. 2. Nash Equilibrium. 3. Mixed Strategies. 4. Backward induction. 5. Re nement. Subgame Perfect Equilibria 6. Repeated Games and the Folk Theorem. 7. Games of Incomplete Information. 8. Some game-theoretic models of oligopoly. References 1. Gibbons, R. (1992). A Primer in Game Theory, Harvester Wheatsheaf. 2. Tirole, J. (1991) The Theory of Industrial Organization, MIT Press. See chapter 11. 3. Fudenberg, D. and J. Tirole (1991). Game Theory, MIT Press. 4. Kreps, D. (1990) Game Theory and Economic Modelling, Clarendon, Oxford. 1
Games inextensiveandstrategic Form The strategic form describes the strategies available to each player and the outcome associated with the choices. The extensive form is useful for dynamic games. Describes the order in which the players move, what their choices are, the information that each player has at each stage and so on. The extensive form is often represented by a game tree. NashEquilibrium Looselyspeaking, a Nash equilibriumisa strategycombination in which each player chooses a best response to the strategies chosen by the other players. The de nition above is given for strategic form games but the idea is general enough to cover extensive form games. In general, we can argue that if there is an obvious way to play the game, this must lead to a Nash equilibrium. Of course, there may exist more than one Nash equilibrium in the game, and hence the existence of a Nash equilibrium does not imply that there is an `obvious way to play the game'. 2
How can we nd Nash Equilibria in simplegames? Consider the following standard coordination game, known as the \battle of the sexes." Firm 1 A B A 2,1-1,-1 Firm 2 B -1,-1 1,2 In essence, we must examine all strategy combinations, and for each one, check to see if the Nash equilibrium conditions are satis ed. a. Start with the strategy combination (A, A). (a1) Look at the payo s from the Firm 1's viewpoint. If Firm 2 chooses B, is A optimal for Firm 1? Yes, because 2> 1: (a2) Now look at the payo s from the Firm 2's viewpoint. If Firm 1 chooses A, is A optimal for Firm 2? Yes, because 1> 1: Since the answer is 'yes' in both (a1) and (a2), (A,A) is a Nash equilibrium. b. Next, consider the strategy combination (A, B). (b1) Look at the payo s from Firm 1's viewpoint. If Firm 2 chooses B, is A optimal for Firm 1? No, because by choosing A it gets -1, and it could do better by choosing B which would fetch 1. For this strategy combination, the Nash equilibrium condition does not hold for the Firm 1. (b2) We could look at this strategy combination from Firm 2's viewpoint, but given that the Nash condition does not hold in (b1), we need not really bother. In short, (A, B) is not a Nash equilibrium. c. Next, consider the strategy combination (B, B). Checking (c1) and (c2), this turns out to be a Nash equilibrium. d. Next, consider the strategy combination (B, A) This is not a Nash equilibrium. In sum, there seem to be two Nash equilibria in this game, namely (A, A) and (B, B). If the game had three strategies for each player, there would be 9 possible strategy combinations for us to check for Nash equilibria. 3
Mixedstrategies Consider the game below, called \matching-pennies." Player 2 Heads Tails Heads 1,-1-1,1 Player 1 Tails -1,1 1,-1 This game seems to have no Nash equilibria, at least in the sense that they have been described thus far. But, in fact it does have a Nash equilibrium, in a manner that might appear curious at rst. The rst stage in the argument is to enlarge the strategy space by constructing probability distributions over the strategy set S. i A mixed strategy s is a probability distribution over the set of (pure) i strategies. In the matching pennies game, a pure strategy might be Heads. A mixed strategy could be Heads with probability 1=3, and Tailswith probability2=3. Notice that a pure-strategy is only a special case of a mixed strategy. A Nash equilibrium can now be de ned in the usual way but using mixed strategies instead of pure strategies. 4
Repeated GamesandtheFolkTheorem Suppose a particular game is played a large number of times. Can we say something about the behaviour of players in such `supergames' that is not obvious in the analysis of the one-shot game? To anticipate the argument, we will try and establish that if the game is repeated a large number of times, we cannot rule out some outcomes that are clearly unlikely in case the game was played just once. First, we need to have an appropriate notion of payo s in the supergame, or more accurately, the relationship between the payo in the supergame and in the one-shot constituent game (now called the stage game). The average payo over the supergame is some aggregate measure of the payo s from the stage games, with later payo s possibly discounted for the lag with which they will become available. The `folk theorems' for repeated games are usually some variant of a simple idea, namely that, if the players are su±ciently patient, then any feasible, individually rational payo combination can be supported as a Nash equilibrium (the two adjectives need some explanation). One conclusion that emerges from this is that outcomes that were ruled out in the oneshot game can be `sustained' in the associated repeated game. The intuition for this might run as follows: if players deviate from say, an agreement to cooperate theycould bepunished by theother(s) in subsequent roundswhich would cause loss of utility in every subsequent period. (How hard they can be punished depends on what their reservation utility is; how costly this future punishment seems to them depends on their discount factor). Fearing this reduction of utility in later rounds (which would reduce the average payo ) breeds compliance or cooperation. This has considerable application in models of say, lender-debtor interaction, product quality, entry deterrence, etc. This idea is quite useful in understanding why cooperation is sustained in the real world but, as such, is extremely inconvenient from a game-theoretic viewpoint: it suggests a large number of possible outcomes, and multiplicity is always abhorred by those who want theory to be a guide to prediction. 5
Somegame-theoreticmodelsofoligopoly We now apply some of the techniques developed so far to model theoligopolistic interaction between rms. Since the aim here is to see how notions such as Nash equilibrium relate to oligopoly, the details of the underlying market are kept as simple as possible. For instance, we readily assume that rms are identical, demand curves are conveniently linear, and so on. More general versions of the examples presented here can easily be generated. Bertrand Equilibrium: Nash equilibrium in prices We rst consider a case with two identical rms, each producing the same homogeneous product. Production is costless. The rms face a standard downward-sloping demand curve, and their individual share of the total market demand depends on the combination of prices charged by them. Put simply, the rm with the lower price captures the entire market; if they charge the same price, they get half the market each. The prices are chosen 'simultaneously'. Let us see how we can model this as a game. Label the rms as rm 1 and 2. The strategy set for each rm can be thought of as the set of non-negative prices, p. The payo function measures the pro t made by a rm for any given strategy combination. This can be written as ¼ (p;p ), for i = 1; 2 andj6=i. In this case, i i j ¼ (p ;p ) =p q 1 1 1 where 8 >q < if p 1 <p 2 q 1 = 0 if p 1 >p 2 >: q=2 if p 1 =p 2. Firm 2's payo function is de ned symmetrically, ¼ 2(p 1;p 2) = p2q 2. Here a Nash equilibrium is a pair of prices (p ;p ) such that 1. ¼ 1(p 1;p 2) ¼ 1(p 1;p 2) 8p 1, and 2. ¼ 2(p 1;p 2) ¼ 1(p 1;p 2) 8p 2. How do we compute the equilibrium? That is, how do we nd the prices that satisfy conditions 1 and 2? Let us look at some candidate solutions. 6
Case 1 p 1>p 2> 0. Does this satisfy 1 and 2? At this con guration of prices, ¼ 1 = 0 and ¼ 2 > 0. (Why?) If rm 1 had chosen some other price, say p 2 ², it could have captured the entire market. Hence the solution could not be of this form. Case 2 p 1 =p 2> 0. This, too, is not an equilibrium. Either rm, if it had chosen a lower price, could have captured a much larger share of the market and made higher pro ts. (Why?) Case 3 p 1>p 2 = 0. This cannot be an equilibrium because rm 2 could have charged a slightly higher price without losing its market share and made strictly greater pro ts. By the process of elimination, we reach the conclusion that the unique Nash equilibrium is given by p =p = 0: The paradoxical feature of this solution is that, in equilibrium, the rms make no (super-normal) pro ts. This seems unrealistic. As an exercise comment on the outcome in the above game if each rm produces at positive, but constant, marginal cost c. What if the marginal cost is not the same for the two rms? Cournot equilibrium: Nashequilibrium in quantities Assume, as in the previous example, that there are two identical rms producing costlessly but now suppose the rms choose, not prices, but quantities to supply to the market. Assuming that the market clears, each rm obtains the market clearing price for each unit of the good supplied. To obtain the payo -function we now need to incorporate the market demand function explicitly. For algebraic simplicity, we use a very simple demand function, q = 120 p: 7
Letq andq be the output levels supplied by rm 1 and 2, respectively. Themarket clearing condition amounts to q +q = q. For any given strategy- combination (q ;q ) the market price is p = 120 (q +q ). Therefore, the payo -function (pro ts) for rm 1 is given as ¼ (q ;q ) =pq = (120 q q )q 1 1 1 and likewise for rm 2. A Nash equilibrium is de ned in the usual way: it is a pair of strategies (namely, quantity choices) (q ;q ) such that 1. ¼ 1(q 1;q 2) ¼ 1(q 1;q 2) 8q 1, and 2. ¼ 2(q 1;q 2) ¼ 1(q 1;q 2) 8q 2. Since in this case the payo functions are di erentiable in the strategy variable, we can obtain the Nash equilibrium by the standard optimization techniques. First, compute the pro t maximizing output choice for each rm taking the rival's output as parametric. Setting @¼ 1 = 0 @q 1 we get, q 1 = 60 :5q 2. This is the \reaction function" for rm 1. Similarly, we can obtain a reaction function for rm 2, q 2 = 60 :5q 1: We then 'solve' the two reaction functions to obtain the Cournot-Nash solution. For this example, it turns out to be q 1 =q 2 = 40. As an exercise generalize the above for the case where the are n rms. Duopoly with sequential moves: the Stackelberg equilibrium Suppose the choice variable is output level as in the Cournot case but the choices are made sequentially. Firm 1 (the `leader') chooses its output level before rm 2 (the `follower') does, and the follower can observe the leader's choice before she makes her output choice. It is not hard to see that the follower's reaction function is as in the previous example but the leader now 8
optimizes taking the follower's reaction function (rather than output level) as given. In other words, the leader maximizes ¼ (q ;q ) =pq = (120 q (60 :5q ))q : 1 1 1 1 1 The Nash equilibrium of this game, also known as the Stackelberg equi- librium, is here given by q = 60; q = 40. CredibleEntry deterrence Consider a situation in which there is one rm in an industry (call it the incumbent) and one or more potential entrants. Entry is costly for the entrants; an entrant will enter only if the post entry pro ts recover the entry cost. The incumbent would prefer to retain the market for itself. The following issues arise this context, depending on how we model the post-entry game. Suppose that there is only one potential entrant and the post-entry game is determined onlybythe actions of the incumbent: theincumbent can either accommodate entry, or to ght it, say, by starting a price war. The price war hurts the incumbent but hurts the potential entrant even more. We will consider some variations along these lines. Theequilibriumnumberof rms Suppose there aren potential entrants, and the post-entry game is given as a symmetric Cournot game among all the rms that are in. Since generally we can compute the Cournot pro ts for a given demand function and given number of rms, if we specify a given level of entry costs, we can determine the number of rms which would each be able to recover their cost of entry. This gives us some sense of the equilibrium number of rms that the market can support. Repeated Games and Collusion 9