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(Cournot's game with many firms) Consider Cournot's game in the case of an arbitrary number \(n\) of firms; retain the assumptions that the inverse demand function takes the form (54.2) and the cost function of each firm \(i\) is \(C_{i}\left(q_{i}\right)=c q_{i}\) for all \(q_{i}\), with \(c<\alpha .\) Find the best response function of each firm and set up the conditions for \(\left(q_{1}^{*}, \ldots, q_{n}^{*}\right)\) to be a Nash equilibrium (see \(\left.(34.3)\right)\), assuming that there is a Nash equilibrium in which all firms' outputs are positive. Solve these equations to find the Nash equilibrium. (For \(n=2\) your answer should be \(\left(\frac{1}{3}(\alpha-c), \frac{1}{3}(\alpha-c)\right)\), the equilibrium found in the previous section. First show that in an equilibrium all firms produce the same output, then solve for that output. If you cannot show that all firms produce the same output, simply assume that they do.) Find the price at which output is sold in a Nash equilibrium and show that this price decreases as \(n\) increases, approaching \(c\) as the number of firms increases without bound. The main idea behind this result does not depend on the assumptions on the inverse demand function and the firms' cost functions. Suppose, more generally, that the inverse demand function is any decreasing function, that each firm's cost function is the same, denoted by \(C\), and that there is a single output, say \(q\), at which the average cost of production \(C(q) / q\) is minimal. In this case, any given total output is produced most efficiently by each firm's producing \(q\), and the lowest price compatible with the firms' not making losses is the minimal value of the average cost. The next exercise asks you to show that in a Nash equilibrium of Cournot's game in which the firms' total output is large relative to \(\underline{q}\), this is the price at which the output is sold.

(Nash equilibrium of Cournot's game with small firms) Suppose that there are infinitely many firms, all of which have the same cost function \(C\). Assume that \(C(0)=0\), and for \(q>0\) the function \(C(q) / q\) has a unique minimizer \(q ;\) denote the minimum of \(C(q) / q\) by \(p\). Assume that the inverse demand function \(\bar{P}\) is decreasing. Show that in any Nash equilibrium the firms' total output \(Q^{*}\) satisfies $$ P\left(Q^{*}+\underline{q}\right) \leq \underline{p} \leq P\left(Q^{*}\right) $$ (That is, the price is at least the minimal value \(p\) of the average cost, but is close enough to this minimum that increasing the total output of the firms by \(q\) would reduce the price to at most \(\underline{p}\).) To establish these inequalities, show that if \(P\left(Q^{*}\right)<\underline{p}\) or \(P\left(Q^{*}+q\right)>p\) then \(Q^{*}\) is not the total output of the firms in a Nash equilibrium, because in each case at least one firm can deviate and increase its profit. 3.1.5 A generalization of Cournot's game: using common property In Cournot's game, the payoff function of each firm \(i\) is \(q_{i} P\left(q_{1}+\cdots+q_{n}\right)-C_{i}\left(q_{i}\right)\). In particular, each firm's payoff depends only on its output and the sum of all the firm's outputs, not on the distribution of the total output among the firms, - and decreases when this sum increases (given that \(P\) is decreasing). That is, the payoff of each firm \(i\) may be written as \(f_{i}\left(q_{i}, q_{1}+\cdots+q_{n}\right)\), where the function \(f_{i}\) is decreasing in its second argument (given the value of its first argument, \(q_{i}\) ). This general payoff function captures many situations in which players compete in using a piece of common property whose value to any one player diminishes as total use increases. The property might be a village green, for example; the higher the total number of sheep grazed there, the less valuable the green is to any given farmer. The first property of a Nash equilibrium in Cournot's model discussed in the previous section applies to this general model: common property is "overused" in a Nash equilibrium in the sense that every player's payoff increases when every player reduces her use of the property from its equilibrium level. For example, all farmers' payoffs increase if each farmer reduces her use of the village green from its equilibrium level: in an equilibrium the green is "overgrazed". The argument is the same as the one illustrated in Figure \(59.1\) in the case of two players, because this argument depends only on the fact that each player's payoff function is smooth and is decreasing in the other player's action. (In Cournot's model, the "common property" that is overused is the demand for the good.)

(Electoral competition with three candidates) Consider a variant of Hotelling's model in which there are three candidates and each candidate has the option of staying out of the race, which she regards as better than losing and worse than tying for first place. Use the following arguments to show that the game has no Nash equilibrium. First, show that there is no Nash equilibrium in which a single candidate enters the race. Second, show that in any Nash equilibrium in which more than one candidate enters, all candidates that enter tie for first place. Third, show that there is no Nash equilibrium in which two candidates enter the race. Fourth, show that there is no Nash equilibrium in which all three candidates enter the race and choose the same position. Finally, show that there is no Nash equilibrium in which all three candidates enter the race, and do not all choose the same position.

(Citizen-candidates) Consider a game in which the players are the citizens. Any citizen may, at some cost \(c>0\), become a candidate. Assume that the only position a citizen can espouse is her favorite position, so that a citizen's only decision is whether to stand as a candidate. After all citizens have (simultaneously) decided whether to become candidates, each citizen votes for her favorite candidate, as in Hotelling's model. Citizens care about the position of the winning candidate; a citizen whose favorite position is \(x\) loses \(\left|x-x^{*}\right|\) if the winning candidate's position is \(x^{*}\). (For any number \(z,|z|\) denotes the absolute value of \(z:|z|=z\) if \(z>0\) and \(|z|=-z\) if \(z<0 .\) ) Winning confers the benefit \(b\). Thus a citizen who becomes a candidate and ties with \(k-1\) other candidates for first place obtains the payoff \(b / k-c\); a citizen with favorite position \(x\) who becomes a candidate and is not one of the candidates tied for first place obtains the payoff \(-\left|x-x^{*}\right|-c\), where \(x^{*}\) is the winner's position; and a citizen with favorite position \(x\) who does not become a candidate obtains the payoff \(-\left|x-x^{*}\right|\), where \(x^{*}\) is the winner's position. Assume that for every position \(x\) there is a citizen for whom \(x\) is the favorite position. Show that if \(b \leq 2 c\) then the game has a Nash equilibrium in which one citizen becomes a candidate. Is there an equilibrium (for any values of \(b\) and \(c\) ) in which two citizens, each with favorite position \(m\), become candidates? Is there an equilibrium in which two citizens with favorite positions different from \(m\) become candidates? Hotelling's model assumes a basic agreement among the voters about the ordering of the positions. For example, if one voter prefers \(x\) to \(y\) to \(z\) and another voter prefers \(y\) to \(z\) to \(x\), no voter prefers \(z\) to \(x\) to \(y\). The next exercise asks you to study a model that does not so restrict the voters' preferences.

(Nash equilibrium of second-price sealed-bid auction) Find a Nash equilibrium of a second-price sealed-bid auction in which player \(n\) obtains the object. Player 2 's bid in this equilibrium exceeds her valuation, and thus may seem a little rash: if player 1 were to increase her bid to any value less than \(v_{1}\), player 2 's payoff would be negative (she would obtain the object at a price greater than her valuation). This property of the action profile does not affect its status as an equilibrium, because in a Nash equilibrium a player does not consider the "risk" that another player will take an action different from her equilibrium action; each player simply chooses an action that is optimal, given the other players' actions. But the property does suggest that the equilibrium is less plausible as the outcome of the auction than the equilibrium in which every player bids her valuation. The same point takes a different form when we interpret the strategic game as a model of events that unfold over time. Under this interpretation, player 2's action \(v_{1}\) means that she will continue bidding until the price reaches \(v_{1}\). If player 1 is sure that player 2 will continue bidding until the price is \(v_{1}\), then player 1 rationally stops bidding when the price reaches \(v_{2}\) (or, indeed, when it reaches any other level at most equal to \(v_{1}\) ). But there is little reason for player 1 to believe that player 2 will in fact stay in the bidding if the price exceeds \(v_{2}\) : player 2 's action is not credible, because if the bidding were to go above \(v_{2}\), player 2 would rationally withdraw. The weakness of the equilibrium is reflected in the fact that player 2 's bid \(v_{1}\) is weakly dominated by the bid \(v_{2} .\) More generally, in a second-price sealed-bid auction (with perfect information), a player's bid equal to her valuation weakly dominates all her other bids. That is, for any bid \(b_{i} \neq v_{i}\), player \(i^{\prime}\) s bid \(v_{i}\) is at least as good as \(b_{i}\), no matter what the other players bid, and is better than \(b_{i}\) for some actions of the other players. (See Definition 45.1.) A player who bids less than her valuation stands not to win in some cases in which she could profit by winning (when the highest of the other bids is between her bid and her valuation), and never stands to gain relative to the situation in which she bids her valuation; a player who bids more than her valuation stands to win in some cases in which she obtains a negative payoff by doing so (when the highest of the remaining bids is between her valuation and her bid), and never stands to gain relative to the situation in which she bids her valuation. The key point is that in a second-price auction, a player who changes her bid does not lower the price she pays, but only possibly changes her status from that of a winner into that of a loser, or vice versa. A precise argument is shown in Figure 84.1, which compares player \(i^{\prime}\) s payoffs to the bid \(v_{i}\) with her payoffs to a bid \(b_{i}

EXERCISE \(85.1\) (First-price sealed-bid auction) Show that in a Nash equilibrium of a first-price sealed-bid auction the two highest bids are the same, one of these bids is submitted by player 1, and the highest bid is at least \(v_{2}\) and at most \(v_{1}\). Show also that any action profile satisfying these conditions is a Nash equilibrium. In any equilibrium in which the winning bid exceeds \(v_{2}\), at least one player's bid exceeds her valuation. As in a second-price sealed-bid auction, such a bid seems "risky", because it would yield the bidder a negative payoff if it were to win. In the equilibrium there is no risk, because the bid does not win; but, as before, the fact that the bid has this property reduces the plausibility of the equilibrium. As in a second-price sealed-bid auction, the potential "riskiness" to player \(i\) of a bid \(b_{i}>v_{i}\) is reflected in the fact that it is weakly dominated by the bid \(v_{i}\), as shown by the following argument. \- If the other players' bids are such that player \(i\) loses when she bids \(b_{i}\), then the outcome is the same 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 her payoff is negative when she bids \(b_{i}\) and zero when she bids \(v_{i}\) (whether or not this bid wins). However, in a first-price auction, unlike a second-price auction, a bid \(b_{i}b_{i}\) because if the other players' highest bid is less than \(b_{i}\) then both \(b_{i}\) and \(b_{i}^{\prime}\) win and \(b_{i}\) yields a lower price. Further, even though the bid \(v_{i}\) weakly dominates higher bids, this bid is itself weakly dominated, by a lower bid! If player \(i\) bids \(v_{i}\) her payoff is 0 regardless of the other players' bids, whereas if she bids less than \(v_{i}\) her payoff is either 0 (if she loses) or positive (if she wins). In summary, in a first-price sealed-bid auction (with perfect information), a player's bid of at least her valuation is weakly dominated, and a bid of less than her valuation is not weakly dominated. An implication of this result is that in every Nash equilibrium of a first- price sealed-bid auction at least one player's action is weakly dominated. However, this property of the equilibria depends on the assumption that a bid may be any number. In the variant of the game in which bids and valuations are restricted to be multiples of some discrete monetary unit \(\epsilon\) (e.g. a cent), an action profile \(\left(v_{2}-\epsilon, v_{2}-\epsilon, b_{3}, \ldots, b_{n}\right)\) for any \(b_{j} \leq v_{j}-\epsilon\) for \(j=3, \ldots, n\) is a Nash equilibrium in which no player's bid is weakly dominated. Further, every equilibrium in which no player's bid is weakly dominated takes this form. When \(\epsilon\) is small, each such equilibrium is close to an equilibrium \(\left(v_{2}, v_{2}, b_{3}, \ldots, b_{n}\right)\) (with \(b_{j} \leq v_{j}\) for \(j=3, \ldots, n)\) of the game with unrestricted bids. On this (somewhat \(a d\) hoc) basis, I select action profiles \(\left(v_{2}, v_{2}, b_{3}, \ldots, b_{n}\right)\) with \(b_{j} \leq v_{j}\) for \(j=3, \ldots, n\) as "distinguished" equilibria of a first-price sealed-bid auction. One conclusion of this analysis is that while both second-price and first- price auctions have many Nash equilibria, yielding a variety of outcomes, their distinguished equilibria yield the same outcome. (Recall that the distinguished equilibrium of a second-price sealed-bid auction is the action profile in which every player bids her valuation.) In every distinguished equilibrium of each game, the object is sold to player 1 at the price \(v_{2} .\) In particular, the auctioneer's revenue is the same in both cases. Thus if we restrict attention to the distinguished equilibria, the two auction forms are "revenue equivalent". The rules are different, but the players' equilibrium bids adjust to the difference and lead to the same outcome: the single Nash equilibrium in which no player's bid is weakly dominated in a second-price auction yields the same outcome as the distinguished equilibria of a first-price auction.

(Third-price auction) Consider a third-price sealed-bid auction, which differs from a first- and a second-price auction only in that the winner (the person who submits the highest bid) pays the third highest price. (Assume that there are at least three bidders.) \(a\). Show that for any player \(i\) the bid of \(v_{i}\) weakly dominates any lower bid, but does not weakly dominate any higher bid. (To show the latter, for any bid \(b_{i}>v_{i}\) find bids for the other players such that player \(i\) is better off bidding \(b_{i}\) than bidding \(v_{i}\).) b. Show that the action profile in which each player bids her valuation is not a Nash equilibrium. c. Find a Nash equilibrium. (There are ones in which every player submits the same bid.) 3.5.4 Variants Uncertain valuations One respect in which the models in this section depart from reality is in the assumption that each bidder is certain of both her own valuation and every other bidder's valuation. In most, if not all, actual auctions, information is surely less perfect. The case in which the players are uncertain about each other's valuations has been thoroughly explored, and is discussed in Section 9.7. The result that a player's bidding her valuation weakly dominates all her other actions in a second-price auction survives when players are uncertain about each other's valuations, as does the revenue- equivalence of first- and second-price auctions under some conditions on the players' preferences. Common valuations In some auctions the main difference between the bidders is not that the value the object differently but that they have different information about its value. For example, the bidders for an oil tract may put similar values on any given amount of oil, but have different information about how much oil is in the tract. Such auctions involve informational considerations that do not arise in the model we have studied in this section; they are studied in Section 9.7.3. Multi-unit auctions In some auctions, like those for Treasury Bills (short- term) government bonds) in the USA, many units of an object are available, and each bidder may value positively more than one unit. In each of the types of auction described below, each bidder submits a bid for each unit of the good. That is, an action is a list of bids \(\left(b^{1}, \ldots, b^{k}\right)\), where \(b^{1}\) is the player's bid for the first unit of the good, \(b^{2}\) is her bid for the second unit, and so on. The player who submits the highest bid for any given unit obtains that unit. The auctions differ in the prices paid by the winners. (The first type of auction generalizes a first-price auction, whereas the next two generalize a second-price auction.) Discriminatory auction The price paid for each unit is the winning bid for that unit. Uniform-price auction The price paid for each unit is the same, equal to the highest rejected bid among all the bids for all units. Vickrey auction A bidder who wins \(k\) objects pays the sum of the \(k\) highest rejected bids submitted by the other bidders. The next exercise asks you to study these auctions when two units of an object are available.

Consider the variant of the War of Attrition in which each player attaches no value to the time spent waiting for the other player to concede, but the object in dispute loses value as time passes. (Think of a rotting animal carcass or a melting ice cream cone.) Assume that the value of the object to each player \(i\) after \(t\) units of time is \(v_{i}-t\) (and the value of a \(50 \%\) chance of obtaining the object is \(\left.\frac{1}{2}\left(v_{i}-t\right)\right) .\) Specify the strategic game that models this sit- uation (take care with the payoff functions). Construct the analogue of Figure \(76.1\), find the players' best response functions, and hence find the Nash equilibria of the game. The War of Attrition is an example of a "game of timing", in which each player's action is a number and each player's payoff depends sensitively on whether her action is greater or less than the other player's action. In many such games, each player's strategic variable is the time at which to act, hence the name "game of timing". The next two exercises are further examples of such games. (In the first the strategic variable is time, whereas in the second it is not.)