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Salop Model of Product Differentiation Consumers are located uniformly on a circle (we normalize its perimeter to 1) and have linear transport costs, T (d) = td. A large number of firms play the following game. In the first stage, they simultaneously decide whether to enter. Those that do get in, are distributed evenly along the circle, maximizing the amount of differentiation. 1 Once in, entrant firms make profits (pi − c)Di − f , where f is the fixed cost of entry and Di is firm i’s demand. Suppose that n firms enter. We look for a symmetric equilibrium in which all firms charge the same price p. Consider now the decision of firm i given that the rest choose p. 2 By setting a price pi, its demand is Di(pi, p) = 2x, where x(0, 1/n) is the location of the consumer that is indifferent between purchasing from i or its neighbour: 1 − x) ⇒ x = 1 p−pi +t/n pi + tx = p + t( n 2 t Profit maximization for firm i yields pi = c + nt . This must then be the equilibrium price. 3 With n firms entering, the equilibrium profits are nt2 − f , which are decreasing in the number of firms. With free entry, n is determined by the zeroprofit condition: n = q t/f . Notice that at the equilibrium level of entry, the price set in stage two, p = c + √ tf is above marginal cost. 4 Example: Dominance in mixed strategies Consider the 2-player normal form game depicted below: Player 1 U M D Player 2 L R 10,1 0,4 4,2 4,3 0,5 10,2 No strict dominance in pure strategies. But if player 1 plays U and D with probability 1 2 each, he has an expected payoff of 5 regardless of player 20s strategy, and this strategy strictly dominates M 5 Correlated equilibria So far, we have assumed that players’ randomisations (when they play mixed strategies) are independent. In the 2×2 coordination games considered earlier, for instance, we can describe the mixed strategy equilibrium as follows: Nature provides private and independently distributed signals (θ1, θ2) ∈ [0, 1] × [0, 1] to the 2 players, and each player i assigns decisions to various possible realisations of θi. 6 Suppose there are also public signals available that both players observe. Let θ ∈ [0, 1] be one such signal. Suppose both decide to 1 and to the game if go the dance if θ ≤ 2 θ > 1 2 (in the ‘battle of partners’ coordination game). Each player’s strategy choice is still random but now their actions are perfectly coordinated, and they always meet. 7 Moreover, these decisions have an equilibrium character; if 1 player follows this decision rule, it is optimal for the other player also to do so. This kind of equilibrium is called a ‘correlated equilibrium’ (due to Aumann) - related to the concept of ‘sunspots’ in Macroeconomics. 8 Proposition: Every game ΓN = [I, {∆(Si)}, {ui(.)}] in which the sets S1, ..., SI have a finite number of elements has a mixed strategy NE (existence) Proposition: A NE exists in game ΓN = [I, {Si}, {ui(.)}], if for all i = 1, ..., I, (i) Si is a nonempty, convex and compact subset of some Euclidean space RM ; (ii) ui(s1, ..., sI ) is continuous in (s1, ..., sI ) and quasiconcave in si. 9 Static games of incomplete information - Bayesian games In a Bayesian game, each player i has a payoff function ui(si, s−i, θi), where θi ∈ Θi is a random variable chosen by nature that is observed only by player i. Joint probability distribution of the θis is given by F (θ1, ...θI ) - common knowledge. Letting Θ = Θ1 × ... × ΘI , a Bayesian game is summarised by [I, {Si}, {ui}, Θ, F (.)] 10 A pure strategy in a Bayesian game for player i is a function si(θi) that gives the player’s strategy choice for each realisation of his type θi. Player i0s pure strategy set Si is the set of all such functions. Player i0s expected payoff for a given profile of pure strategies for the I players (s1(.), ..., sI (.)) is then given by: ũi(si(.), ....sI (.)) = Eθ [ui(s1(θ1), ...sI (θI ), θi)] 11 Definition: A (pure strategy) Bayesian Nash equilibrium (BNE) for the Bayesian game [I, {Si}, {ui}, Θ, F (.)] is a profile of decision rules (s1(.), ..., sI (.)) that constitutes a Nash equlibrium of game ΓN = [I, {Si}, {ũi}]. That is, for every i = 1, ...I, ũi(si(.), s−i(.)) ≥ ũi(s0i(.), s−i(.)) ∀s0i(.) ∈ Si. 12 Proposition: A profile of decision rules (s1(.), ...sI (.)) is a BNE for the Bayesian game [I, {Si}, {ui}, Θ, F (.)] iff, ∀ i and ∀ θ¯i ∈ Θi occurring with positive probability, Eθ−i [ui(si(θ¯i), s−i(θ−i), θ¯i|θ¯i] ≥ Eθ−i [ui(s0i, s−i(θ−i), θ¯i|θ¯i] ∀ s0i ∈ Si, where the expectation is taken over realisations of the other players’ random variables, conditional on player i0s realisation of his signal θ¯i. 13 i.e., we can think of each type of player i as being a separate player who maximises his payoff given his conditional probability distribution over the strategy choices of his rivals. Player 1 Player 1 Don’t confess Confess Player 2 (µ) Don’t Confess confess 0,-2 -10,-1 -1,-10 -5,-5 Don’t confess Confess Player 2 (1 − µ) Don’t Confess confess 0,-2 -10,-6 -1,-10 -5,-11 With probability µ, prisoner 2 has preferences as in 1st matrix (Type I), and with probability (1 − µ), he’s the type who hates to snitch - he gets a psychic penalty equal to 6 years in prison for confessing (Type II). 14 Player 2 now has 4 possible pure strategies: (confess if type I, confess if type II); (confess if type I, don’t confess if type II); (don’t confess if type I, confess if type II); (don’t confess if type I, don’t confess if type II) Player 1 does not observe player 2’s type, so a pure strategy for player 1 is simple {confess, don’t confess} 15 To solve for the BNE of the game, note that type I player must play ‘confess’ with probability 1 - that is his dominant strategy. Likewise, type II has a dominant strategy ‘don’t confess’. Given this, player 10s best response is to play 1 and to play ‘confess’ ‘don’t confess’ if µ < 6 1. . He’s indifferent if µ = if µ > 1 6 6 16 First-price sealed bid auction We have 2 bidders i = 1, 2. Bidder i has valuation vi for the good. The 2 bidders’ valuations are independently and uniformly distributed on [0, 1]. Bids bi ≥ 0. Bidders simultaneously submit bids. Higher bidder wins the good and pays the price she bid; other bidder gets nothing. In case of a tie, winner determined by flip of a coin. Bidders are risk-neutral. All of this is common knowledge. 17 Here, Ai = [0, ∞); Θ = [0, 1] × [0, 1]; vi − bi, bi 0, bi < ui(b1, b2; v1, v2) = (vi−bi) , b i 2 > bj bj = bj A strategy for player i is a function bi(vi). In a BNE, player 10s strategy b1(v1) is a BR to player 20s strategy b2(v2), and vice versa. The pair of strategies (b(v1), b(v2)) is a BNE if for each bi in [0, 1], bi(vi) solves: 1 (v −b )P r{b = maxbi (vi−bi)P r{bi > bj (vj )}+ 2 i i i bj (vj )} 18 We simplify the problem by looking for a linear equilibrium α1 + β1v1 and α2 + β2v2. Suppose player j adopts the strategy αj + βj vj . For given vi, player i0s BR solves: maxbi (vi − bi)P r(bi > αj + βj vj ) We have αj ≤ bi ≤ αj + βj , for a sensible strategy! So, P r(bi > αj + βj vj ) = P r{vj < bi −αj = βj bi −αj βj } 19 Player i0s BR is therefore: vi +αj 2 , v i ≥ αj bi(vi) = α ,v < α j i j Since we are looking for a linear equilibrium, we rule out cases where 0 < αj < 1, looking instead at ranges where αj ≤ 0 and αj ≥ 1. But αj ≥ 1 cannot be an equilibrium - since it is optimal for a higher type to bid at least as much as a lower type’s optimal bid, we have cj ≥ 0. 20 But then αj ≥ 1 would imply bj (vj ) ≥ vj , which cannot be optimal. So, for a linear equilibrium, we must have αj ≤ 0, which ⇔ bi(vi) = vi +αj 2 , so that α αi = 2j and ci = 1 2. Using symmetry to derive j 0s BR, and combining the results, we get αi = αj = 0 and 1. βi = βj = 2 v The BNE of this game is { v2i , 2j } 21