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Chapter 3: Problem 29

(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.

Short Answer

Expert verified
Bidding one's valuation weakly dominates lower bids but not higher bids. Each player bidding their valuation is not a Nash equilibrium. All players bidding 0 is a Nash equilibrium.

Step by step solution

01

- Understand Nash Equilibrium and Weak Dominance

Review the concepts of Nash equilibrium and weak dominance. A strategy weakly dominates another if it is at least as good in all possible scenarios and strictly better in at least one scenario. Nash equilibrium occurs when no player can benefit by unilaterally changing their strategy.
02

- Analyze Third-Price Auction

Recognize that in a third-price auction, the highest bidder wins but pays the third-highest price. The focus will be on showing how bidding one's valuation is weakly dominant in some cases and not in others.
03

Step 3a - Bidding Your Valuation Weakly Dominates Lower Bids

Consider a player with valuation \(v_i\). If they bid \(v_i\), they will win if \(v_i\) is the highest bid and pay the third-highest price. If they bid lower than \(v_i\), there is a chance they lose when they could have won. Therefore, bidding \(v_i\) is at least as good as bidding lower.
04

Step 3b - Bidding Your Valuation Does Not Weakly Dominate Higher Bids

To demonstrate that bidding \(b_i > v_i\) can be better: suppose the other two highest bids are both slightly above \(v_i\) but below \(b_i\), say \( v_i + \epsilon_1 \text{and} v_i + \epsilon_2 \) with \( \epsilon_1, \epsilon_2 > 0 \) and \( \epsilon_2 > \epsilon_1 \). Bidding \(b_i\) would win and pay \(v_i + \epsilon_1\), whereas bidding \(v_i\) would lose.
05

- Show Bidding Your Valuation is Not Nash Equilibrium

In a Nash equilibrium, no player can improve by changing their strategy unilaterally. If each player bids their valuation, one player could potentially increase their bid slightly to avoid paying the third-highest price while still winning. Thus, bidding your valuations is not a Nash equilibrium.
06

- Find the Nash Equilibrium

Consider a scenario where all players bid a sufficiently low amount, say 0. Since they are indifferent to winning (they would pay 0), no player gains from unilaterally deviating. Thus, all players bidding 0 constitutes a Nash equilibrium.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Game Theory
Game theory is a field of study that examines strategic interactions between individuals or groups, where the outcome for each participant depends on the actions of all. In auctions, game theory helps analyze how bidders make decisions to maximize their payoffs.
In our context of a third-price auction:
  • Individuals try to outbid their opponents while considering the third highest price will be paid by the winner.
  • The strategies chosen by one bidder can affect both their success and the payoffs of other bidders.
Understanding game theory is crucial to grasp auction behaviors as bidders need to strategically plan their bids while considering others' moves.
Nash Equilibrium
Nash equilibrium is a concept in game theory where no player can improve their outcome by unilaterally changing their strategy. Essentially:
  • Each player's strategy is the best response to the strategies chosen by the others.
  • No single player can benefit by deviating on their own.
In a third-price auction:
While a strategy where each player bids their own valuation seems logical, it fails as a Nash equilibrium because:
  • A player can gain by bidding slightly higher, ensuring they win but pay the third-highest price.
  • This potential improvement by just one player disrupts the equilibrium.
Therefore, we need to find other bid strategies that provide mutual best responses among all players.
Weak Dominance
In game theory, a strategy weakly dominates another if it provides at least the same payoff in all scenarios and a better payoff in at least one scenario. For a third-price auction:
Bidding your true valuation weakly dominates bidding lower as:
  • If you bid your valuation, you may win and pay the third-highest price.
  • If you bid lower, you risk losing when you could have possibly won.
However, bidding higher than your valuation does not weakly dominate:
It might sometimes result in lower costs if the highest bids from others are above your valuation, but still unprofitably high compared to what you would ideally pay.
Auction Theory
Auction theory explores how different auction formats influence bidding behaviors and outcomes. The third-price auction is unique because:
  • The highest bidder wins but pays only the third-highest price.
  • This leads to strategic considerations different from first- and second-price auctions.
In a third-price auction:
A bidder has to balance their desire to win with the optimal price they are willing to pay. Unlike straightforward highest-bid wins formats, the third-price auction complicates how one values their strategy against others' bids.
Bidding Strategy
In any auction, the strategy involves deciding how much to bid to maximize your payoff without paying more than necessary.
Specifically for a third-price auction, optimal bidding strategies can include:
  • Assessing the value of the item being auctioned accurately.
  • Considering the possible bids of other participants.
  • Deciding whether to bid your valuation, slightly higher, or adopt a different strategy to balance winning and payment optimization.
Effective strategies might involve:
Bidding closer to true valuations but also considering potential higher or lower bids to impact final costs and winning chances.

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Most popular questions from this chapter

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.

(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 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.)

(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.

(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.
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