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Nash Equilibrium: Illustrations

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Cournot’s Model of Oligopoly Single good produced by n firms Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing If firms’ total output is Q then market price is P(Q), where P is nonincreasing Profit of firm i, as a function of all the firms’ outputs:

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Cournot’s Model of Oligopoly Strategic game: players: firms each firm’s set of actions: set of all possible outputs each firm’s preferences are represented by its profit

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Example: Duopoly two firms Inverse demand: constant unit cost: C i (q i ) = cq i, where c <

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Example: Duopoly

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Recall for a perfectly competitive firm, P=MC, so – Q = c, or Q = – c. Recall for a monopolist, MR=MC, so – 2Q = c, or Q = ( – c)/2.

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Example: Duopoly

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Best response function is: Same for firm 2: b 2 (q) = b 1 (q) for all q.

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Example: Duopoly

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Nash equilibrium: Pair (q* 1, q* 2 ) of outputs such that each firm’s action is a best response to the other firm’s action or q* 1 = b 1 (q* 2 ) and q* 2 = b 2 (q* 1 ) Solution: q 1 = ( - c - q 2 )/2 and q 2 = ( - c - q 1 )/2 q* 1 = q* 2 = ( - c)/3

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Example: Duopoly

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Conclusion: Game has unique Nash equilibrium: (q* 1, q* 2 ) = (( - c)/3, ( - c)/3) At equilibrium, each firm’s profit is = ( - c) 2 )/9 Total output 2/3( - c) lies between monopoly output ( - c)/2 and competitive output - c.

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Cournot’s Model of Oligopoly Comparison of Nash equilibrium with collusive outcomes. Can both firms do better than the Cournot Nash? Yes! (if they can collude) –Exercise 60.2 in Osborne Dependence of Nash equilibrium on number of firms. What happens to the Nash quantity and price as the number of firms grows? –Exercise 61.1 in Osborne

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Bertrand’s Model of Oligopoly Strategic variable price rather than output. Single good produced by n firms Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing If price is p, demand is D(p) Consumers buy from firm with lowest price Firms produce what is demanded

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Bertrand’s Model of Oligopoly Strategic game: players: firms each firm’s set of actions: set of all possible prices each firm’s preferences are represented by its profit

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Example: Duopoly 2 firms C i (q i ) = cq i for i = 1, 2 D(p) = - p

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Example: Duopoly Nash Equilibrium (p 1, p 2 ) = (c, c) If each firm charges a price of c then the other firm can do no better than charge a price of c also (if it raises its price it sells no output, while if it lowers its price it makes a loss), so (c, c) is a Nash equilibrium.

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Example: Duopoly No other pair (p 1, p 2 ) is a Nash equilibrium since If p i < c then the firm whose price is lowest (or either firm, if the prices are the same) can increase its profit (to zero) by raising its price to c If p i = c and p j > c then firm i is better off increasing its price slightly if p i ≥ p j > c then firm i can increase its profit by lowering p i to some price between c and p j (e.g. to slightly below p j if D(p j ) > 0 or to p m if D(p j ) = 0).

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Example: Duopoly 45 p1p1 p2p2 c c pmpm pmpm P 2 = BR 2 (p 1 ) P 1 = BR 1 (p 2 )

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Hotelling’s Model of Electoral Competition Several candidates run for political office Each candidate chooses a policy position Each citizen, who has preferences over policy positions, votes for one of the candidates Candidate who obtains the most votes wins.

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Hotelling’s Model of Electoral Competition Strategic game: Players: candidates Set of actions of each candidate: set of possible positions Each candidate gets the votes of all citizens who prefer her position to the other candidates’ positions; each candidate prefers to win than to tie than to lose. Note: Citizens are not players in this game.

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Example Two candidates Set of possible positions is a (one-dimensional) interval. Each voter has a single favorite position, on each side of which her distaste for other positions increases equally. Unique median favorite position m among the voters: the favorite positions of half of the voters are at most m, and the favorite positions of the other half of the voters are at least m.

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Example Direct argument for Nash equilibrium (m, m) is an equilibrium: if either candidate chooses a different position she loses. No other pair of positions is a Nash equilibrium: If one candidate loses then she can do better by moving to m (where she either wins or ties for first place) If the candidates tie (because their positions are either the same or symmetric about m), then either candidate can do better by moving to m, where she wins.

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The War of Attrition Two parties involved in a costly dispute E.g. two animals fighting over prey Each animal chooses time at which it intends to give up Once an animal has given up, the other obtains all the prey If both animals give up at the same time then they split the prey equally. Fighting is costly: each animal prefers as short a fight as possible. Also a model of bargaining between humans.

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The War of Attrition Strategic game players: the two parties each player’s set of actions is [0,∞) (set of possible concession times) player i’s preferences are represented by payoff function

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The War of Attrition If t 1 = t 2 then either player can increase her payoff by conceding slightly later (in which case she obtains the object for sure, rather than getting it with probability 1/2 ). If 0 < t i < t j then player i can increase her payoff by conceding at 0. If 0 = t i 0) by conceding slightly after t j.

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The War of Attrition Thus there is no Nash equilibrium in which t 1 = t 2, 0 < t i < t j, or 0 = t i < t j < v i (for i = 1 and j = 2, or i = 2 and j = 1). The remaining possibility is that 0 = t i < t j and t j ≥ v i for i = 1 and j = 2, or i = 2 and j = 1. In this case player i’s payoff is 0, while if she concedes later her payoff is negative; player j’s payoff is v j, her highest possible payoff in the game.

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The War of Attrition In no equilibrium is there any fight There is an equilibrium in which either player concedes first, regardless of the sizes of the valuations. Equilibria are asymmetric, even when v 1 = v 2, in which case the game is symmetric.

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Auctions

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Second-Price Sealed-Bid Auction Assume every bidder knows her own valuation and every other bidder’s valuation for the good being sold Model each person decides, before auction begins, maximum amount she is willing to bid person who bids most wins person who wins pays the second highest bid.

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Second-Price Sealed-Bid Auction Strategic game: players: bidders set of actions of each player: set of possible bids (nonnegative numbers) preferences of player i: represented by a payoff function that gives player i v i - p if she wins (where v i is her valuation and p is the second- highest bid) and 0 otherwise.

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Second-Price Sealed-Bid Auction Simple (but arbitrary) tie-breaking rule: number players 1,..., n and make the winner the player with the lowest number among those that submit the highest bid. Assume that v 1 > v 2 > · · · > v n.

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Nash equilibria of second-price sealed- bid auction One Nash equilibrium (b 1,..., b n ) = (v 1,..., v n ) Outcome: player 1 obtains the object at price v 2 ; her payoff is v 1 - v 2 and every other player’s payoff is zero.

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Nash equilibria of second-price sealed- bid auction Reason: Player 1: –If she changes her bid to some x ≥ b 2 the outcome does not change (remember she pays the second highest bid) –If she lowers her bid below b 2 she loses and gets a payoff of 0 (instead of v 1 - b 2 > 0).

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Nash equilibria of second-price sealed- bid auction Players 2,..., n: –If she lowers her bid she still loses –If she raises her bid to x ≤ b 1 she still loses –If she raises her bid above b 1 she wins, but gets a payoff v i - v 1 < 0.

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Nash equilibria of second-price sealed- bid auction Another Nash equilibrium (v 1, 0,..., 0) is also a Nash equilibrium Outcome: player 1 obtains the object at price 0; her payoff is v 1 and every other player’s payoff is zero.

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Nash equilibria of second-price sealed- bid auction Reason: Player 1: –any change in her bid has no effect on the outcome Players 2,..., n: –if she raises her bid to x v 1 she still loses –if she raises her bid above v 1 she wins, but gets a negative payoff v i - v 1.

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Nash equilibria of second-price sealed- bid auction For each player i the action v i weakly dominates all her other actions That is: player i can do no better than bid v i no matter what the other players bid.

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Nash equilibria of second-price sealed- bid auction b ’ i < v i b ’ i = v i b ’’ i > v i IPayoff if b ^ -i b ’ i v i - b ^ -i + v i - b ^ -i + v i - b ^ -i + IIPayoff if b ’ i < b ^ -i v i 0v i - b ^ -i + v i - b ^ -i + IIIPayoff if v i < b ^ -i b ’’ i 00v i - b ^ -i - IVPayoff if b ’’ i < b ^ -i 000

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First-Price Sealed-Bid Auction Strategic game: players: bidders actions of each player: set of possible bids (nonnegative numbers) preferences of player i: represented by a payoff function that gives player i v i - p if she wins (where v i is her valuation and p is her bid) and 0 otherwise.

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Nash Equilibria of First-Price Sealed-Bid Auction (b 1,..., b n ) = (v 2, v 2, v 3,..., v n ) is a Nash equilibrium Reason: If player 1 raises her bid she still wins, but pays a higher price and hence obtains a lower payoff. If player 1 lowers her bid then she loses, and obtains the payoff of 0. If any other player changes her bid to any price at most equal to v 2 the outcome does not change. If she raises her bid above v 2 she wins, but obtains a negative payoff.

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Nash Equilibria of First-Price Sealed- Bid Auction Property of all equilibria In all equilibria the object is obtained by the player who values it most highly (player 1) Argument: If player i ≠ 1 obtains the object then we must have b i > b 1. But there is no equilibrium in which b i > b 1 : If b i > v 2 then i’s payoff is negative, so she can do better by reducing her bid to 0 If b i ≤ v 2 then player 1 can increase her payoff from 0 to v 1 - b i by bidding b i.

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First-Price Sealed-Bid Auction As in a second-price auction, any player i’s action of bidding b i > v i is weakly dominated by the action of bidding v i : –If the other players’ bids are such that player i loses when she bids b i, then it makes no difference to her whether she bids b i or v i –If the other players’ bids are such that player i wins when she bids b i, then she gets a negative payoff bidding b i and a payoff of 0 when she bids v i

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First-Price Sealed-Bid Auction Unlike a second-price auction, a bid b i < v i of player i is NOT weakly dominated by any bid. –It is not weakly dominated by a bid b i ’ < b i –Neither by a bid b i ’ > b i

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Revenue Equivalence The price at which the object is sold, and hence the auctioneer’s revenue, is the same in the equilibrium (v 1, v 2,..., v n ) of the second-price auction as it is in the equilibrium (v 2, v 2, v 3,..., v n ) of the first-price auction.

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