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Chapter 3: Problem 25

(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}

Short Answer

Expert verified
Each player bids their valuation \(v_{i}\) because it weakly dominates all other bids, forming a Nash equilibrium.

Step by step solution

01

Identify Player Valuations

Let each player have a valuation for the object. Assume player 1鈥檚 valuation is denoted by \(v_{1}\) and player 2's valuation is denoted by \(v_{2}\).
02

Understanding the Rules

In a second-price sealed-bid auction, the player who submits the highest bid wins the object but pays the amount of the second-highest bid.
03

Analyze Player 1's Optimal Bid

Player 1 should bid \(v_{1}\). Any bid lower than her valuation risks losing the auction when she could profitably win. Any bid higher than her valuation risks overpaying.
04

Analyze Player 2's Optimal Bid

Similarly, player 2 should bid \(v_{2}\). Bidding less than \(v_{2}\) risks not winning when she could win beneficially. Bidding more than \(v_{2}\) risks winning at a loss.
05

Player 2 Bids Exceeds Her Valuation

If player 2 bids \(v_{1}\) exceeds her own valuation \(v_{2}\), it suggests irrational behavior not in equilibrium, as she may win at a price above her valuation.
06

Nash Equilibrium and Weak Dominance

A Nash Equilibrium strategy profile is one where no player can benefit by unilaterally changing their strategy. Bidding each player鈥檚 own valuation \(v_{i}\) weakly dominates any other bid.
07

Conclusion on Nash Equilibria

In the second-price sealed-bid auction, the equilibrium where every player's bid equals their valuation \(\left(b_{1}, \ldots, b_{n}\right)=\left(v_{1}, \ldots, v_{n}\right)\) is distinguished as each player's action weakly dominates any other action.

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Key Concepts

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

second-price sealed-bid auction
In a second-price sealed-bid auction, each player submits a bid without knowing the bids of the other players. The player with the highest bid wins the object being auctioned, but the price paid is equal to the second-highest bid.
This type of auction has an interesting feature: the winning player does not pay their own bid, but rather the second highest bid. This ensures players bid their true valuation without fear of overpaying if they win.
For instance, if Player 1 values the object at 100 units and Player 2 values it at 70 units, one possible outcome is Player 1 wins the object but only pays 70 units if they both bid truthfully. This bidding strategy is considered optimal due to the nature of the second-price rule.
player valuation in game theory
Player valuation in game theory refers to the value each player assigns to an object in a strategic setting. In auctions, each player has a unique valuation they are willing to pay for the item. These valuations are typically represented in terms of numbers like \(v_1\) for Player 1 and \(v_2\) for Player 2.
This value is crucial because it drives the bidding strategy. A player鈥檚 valuation dictates the maximum amount they are willing to bid.
In the context of our second-price sealed-bid auction, players should ideally bid their true valuation. If they bid less, they might lose the auction even if their actual willingness to pay is higher than some other player's bid. If they bid more than their valuation, they may end up winning the auction but at a cost higher than what they value the object, leading to a negative payoff.
weakly dominated strategies
In game theory, a strategy is weakly dominated if there exists another strategy that gives a player a payoff at least as good for all possible actions of the other players, and strictly better for some actions of the others.
In the context of the second-price sealed-bid auction, bidding one鈥檚 true valuation weakly dominates any other bids. This is because:
  • If a player bids lower than their valuation, they risk losing despite being willing to pay more.
  • If a player bids higher than their valuation, they might win but pay more than what the object is worth to them, resulting in a negative payoff.

Therefore, bidding one's own valuation ensures the player avoids the risk of overpaying while also maximizing the chance of winning if it鈥檚 beneficial.
optimal bidding strategy
The optimal bidding strategy in a second-price sealed-bid auction is simple: bid your true valuation. This strategy is optimal for several reasons:
  • It ensures the player does not overpay as they only pay the second-highest bid.
  • If the bid equals the player's valuation, they only win when it's beneficial (when their valuation is greater than the second-highest bid).
  • This strategy prevents the player from any regrets or losses associated with misjudged bids.

To sum up, the optimal bidding strategy leverages the unique nature of the second-price rule, protecting the bidder from the risks of overpaying or strategically losing opportunities.

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Most popular questions from this chapter

There is a finite number of positions and a finite, odd, number of voters. For any positions \(x\) and \(y\), each voter either prefers \(x\) to \(y\) or prefers \(y\) to \(x\). (No voter regards any two positions as equally desirable.) We say that a position \(x^{*}\) is a Condorcet winner if for every position \(y\) different from \(x^{*}\), a majority of voters prefer \(x^{*}\) to \(y\). a. Show that for any configuration of preferences there is at most one Condorcet winner. b. Give an example in which no Condorcet winner exists. (Suppose there are three positions \((x, y\), and \(z)\) and three voters. Assume that voter 1 prefers \(x\) to \(y\) to \(z\). Construct preferences for the other two voters such that one voter prefers \(x\) to \(y\) and the other prefers \(y\) to \(x\), one prefers \(x\) to \(z\) and the other prefers \(z\) to \(x\), and one prefers \(y\) to \(z\) and the other prefers \(z\) to \(y .\) The preferences you construct must, of course, satisfy the condition that a voter who prefers \(a\) to \(b\) and \(b\) to \(c\) also prefers \(a\) to \(c\), where \(a, b\), and \(c\) are any positions.) c. Consider the strategic game in which two candidates simultaneously choose positions, as in Hotelling's model. If the candidates choose different positions, each voter endorses the candidate whose position she prefers, and the candidate who receives the most votes wins. If the candidates choose the same position, they tie. Show that this game has a unique Nash equilibrium if the voters' preferences are such that there is a Condorcet winner, and has no Nash equilibrium if the voters' preferences are such that there is no Condorcet winner.

(Timing product release) Two firms are developing competing products for a market of fixed size. The longer a firm spends on development, the better its product. But the first firm to release its product has an advantage: the customers it obtains will not subsequently switch to its rival. (Once a person starts using a product, the cost of switching to an alternative, even one significantly better, is too high to make a switch worthwhile.) A firm that releases its product first, at time \(t\), captures the share \(h(t)\) of the market, where \(h\) is a function that increases from time 0 to time \(T\), with \(h(0)=0\) and \(h(T)=1 .\) The remaining market share is left for the other firm. If the firms release their products at the same time, each obtains half of the market. Each firm wishes to obtain the highest possible market share. Model this situation as a strategic game and find its Nash equilibrium (equilibria?). (When finding firm \(i\) 's best response to firm \(j\) 's release time \(t_{j}\), there are three cases: that in which \(h\left(t_{j}\right)<\frac{1}{2}\) (firm \(j\) gets less than half of the market if it is the first to release its product), that in which \(h\left(t_{j}\right)=\frac{1}{2}\), and that in which \(\left.h\left(t_{j}\right)>\frac{1}{2} .\right)\)

(Multi-unit auctions) Two units of an object are available. There are \(n\) bidders. Bidder \(i\) values the first unit that she obtains at \(v_{i}\) and the second unit at \(w_{i}\), where \(v_{i}>w_{i}>0\). Each bidder submits two bids; the two highest bids win. Retain the tie-breaking rule in the text. Show that in discriminatory and uniform-price auctions, player \(i^{\prime}\) s action of bidding \(v_{i}\) and \(w_{i}\) does not dominate all her other actions, whereas in a Vickrey auction it does. (In the case of a Vickrey auction, consider separately the cases in which the other players' bids are such that player \(i\) wins no units, one unit, and two units when her bids are \(v_{i}\) and \(w_{i}\).) Goods for which the demand exceeds the supply at the going price are sometimes sold to the people who are willing to wait longest in line. We can model such situations as multi-unit auctions in which each person's bid is the amount of time she is willing to wait.

(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.

(Bertrand's duopoly game with constant unit cost) Consider the extent to which the analysis depends upon the demand function \(D\) taking the specific form \(D(p)=\alpha-p\). Suppose that \(D\) is any function for which \(D(p) \geq 0\) for all \(p\) and there exists \(\bar{p}>c\) such that \(D(p)>0\) for all \(p \leq \bar{p} .\) Is \((c, c)\) still a Nash equilibrium? Is it still the only Nash equilibrium?
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