/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 (Nash equilibrium of second-pric... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(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.

(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.

(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.)

(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.)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.