91Ó°ÊÓ

(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.

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.

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 91Ó°ÊÓ!

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.