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Chapter 3: Problem 86

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.)

Short Answer

Expert verified
In a third-price sealed-bid auction, bidding one's value is not Nash equilibrium since deviation better-off feasible. Set equal same for Nash findings.

Step by step solution

01

Understanding the Auction Setup

In a third-price sealed-bid auction, the winner is the highest bidder but they pay the third-highest bid price. There are at least three bidders.
02

Dominance of bid equals valuation ( v_i ) over any lower bid

To show that bidding v_i weakly dominates any lower bid b_i , consider cases where b_i < v_i : 1. If another player bids between b_i and v_i , then player i wins and pays b_i if b_i wins, but loses if v_i is bid. 2. If another player bids more than v_i , player i loses either way v_i or b_i ( b_i <= v_i). Hence v_i better.
03

No Dominance Over Any Higher Bid

A higher bid b_{i}>v_{i} can sometimes be better. Set other players' bids as follows to show this: Suppose b_i exceeds the third-highest bid but loses otherwise. If b_i > wins but not vi, hence there are cases where b_i > v_i is better than vi.
04

Profile in which all players bid their value is not Nash equilibrium

Assume all players bid their values (true) in apparent Nash: player deviation strategy( bid less than their had) if lower otherwise matching equal; one player better off shows others also follow though losses can eventuate.
05

Finding Nash equilibrium

For Nash equilibrium , assume that all players bid a fixed amount b* ~ threshold below tie expected as all same bid hence per nash equilibrium equal effective price i.e b ~n = b1.

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Key Concepts

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

Dominance Strategy
In a third-price sealed-bid auction, a dominance strategy involves figuring out the best approach for bidding that consistently gives better or equal outcomes regardless of the other players' bids. Here, the primary thing to understand is that bidding your true valuation, denoted as \(v_i\), weakly dominates any lower bid \(b_i\). This means that bidding \(v_i\) is always at least as good as bidding any amount lower than \(v_i\), and sometimes even better.

To see why, consider what happens when you bid lower than your actual valuation:
  • If another player bids an amount between your lower bid \(b_i\) and your true value \(v_i\), you would have won by bidding \(v_i\) but lose with \(b_i\).
  • If other bids are higher than your true valuation \(v_i\), then regardless of whether you bid \(b_i\) or \(v_i\), you still lose. So, in this case, bidding \(v_i\) and \(b_i\) have the same result.
Given these scenarios, bidding your true valuation \(v_i\) ensures you're never worse off and might be better off compared to bidding lower.

However, if you consider higher bidding, let's say \(b_i > v_i\), there are situations where bidding higher may lead to better results, unlike bidding lower. Imagine another player's bid is just under \(v_i\). By bidding slightly more, say \(b_i\), you may win, while bidding exactly \(v_i\), you might lose.
Nash Equilibrium
To explore the Nash equilibrium in a third-price sealed-bid auction, we need to understand that a Nash equilibrium arises when no player can benefit from unilaterally changing their strategy. However, when every player bids their true valuation, this scenario does not create a Nash equilibrium.

For example, if each player is bidding their valuation and one player thinks of deviating by bidding slightly lower, they might believe they could avoid overpaying if they win. This reasoning means that the payoff might improve for this player, suggesting a deviation from the so-called equilibrium strategy.

When each player鈥檚 strategy is to bid exactly their valuations, it's not stable. Each player would constantly see opportunities to gain slight advantages by deviating from their currently true valuations. Consequently, this setup fails to be a Nash Equilibrium because individual players find incentives to change their strategies.
Bid Valuation
Bid valuation determines how each player evaluates their bids compared to their real valuations. In the context of a third-price sealed-bid auction, a player's bid should carefully reflect their true valuation to ensure they are not overpaying or underbidding.

Each bid placed in this auction carries significant implications:
  • Bids below valuation can lead to losing the auction even when the price is affordable.
  • Bids above valuation might unnecessarily increase the payment and lead to potential overspending.
  • Bidding exactly your valuation offers a balance but can sometimes still lead to losing even if justifiable considering strategic contexts.
Ultimately, the goal is to understand that while bidding the true valuation \(v_i\) is dominant over lower bids and might seem a good strategy, it might not form a Nash Equilibrium. Players need to factor in other players鈥� potential strategies and bids while deciding on their own optimal bidding strategy to establish a real equilibrium state where nobody wants to deviate for a better outcome.

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Most popular questions from this chapter

In the variant of Hotelling's model that captures competing firms' choices of product characteristics, show that when there are two firms the unique Nash equilibrium is \((m, m)\) (both firms offer the consumers' median favorite product) and when there are three firms there is no Nash equilibrium. (Start by arguing that when there are two firms whose products differ, either firm is better off making its product more similar to that of its rival.)

(Electoral competition for more general preferences) There is a finite number of positions and a finite, odd, number of voters. For any positions \(x\) ind \(y\), 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 very 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. A variant of Hotelling's model of electoral competition can be used to analyze he choices of product characteristics by competing firms in situations in which lext exercise you are asked to show that the Nash equilibria of this game in the fase of two or three firms are the same as those in Hotelling's model of electoral competition.

(Nash equilibrium of first-price sealed-bid auction) Show that \(\left(b_{1}\right.\), \(\left.\ldots, b_{n}\right)=\left(v_{2}, v_{2}, v_{3}, \ldots, v_{n}\right)\) is a Nash equilibrium of a first-price sealed-bid auction. A first-price sealed-bid auction has many other equilibria, but in all equilibria the winner is the player who values the object most highly (player 1 ), by the following argument. In any action profile \(\left(b_{1}, \ldots, b_{n}\right)\) in which some player \(i \neq 1\) wins, we have \(b_{i}>b_{1}\). If \(b_{i}>v_{2}\) then \(i\) 's payoff is negative, so that she can do better by reducing her bid to 0 ; if \(b_{i} \leq v_{2}\) then player 1 can increase her payoff from 0 to \(v_{1}-b_{i}\) by bidding \(b_{i}\), in which case she wins. Thus no such action profile is a Nash equilibrium.

(Electoral competition with asymmetric voters' preferences) Consider a variant of Hotelling's model in which voters's preferences are asymmetric. Specifically, suppose that each voter cares twice as much about policy differences to the left of her favorite position than about policy differences to the right of her favorite position. How does this affect the Nash equilibrium? In the model considered so far, no candidate has the option of staying out of the race. Suppose that we give each candidate this option; assume that it is better than losing and worse than tying for first place. Then the Nash equilibrium remains as before: both players enter the race and choose the position \(m\). The direct argument differs from the one before only in that in addition we need to check that there is no equilibrium in which one or both of the candidates stays out of the race. If one candidate stays out then, given the other candidate's position, she can enter and either win outright or tie for first place. If both candidates stay out, then either candidate can enter and win outright. The next exercise asks you to consider the Nash equilibria of this variant of the model when there are three candidates.

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 \(i^{\prime}\) 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 \(h\left(t_{j}\right)>\frac{1}{2}\).)
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