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Chapter 27: Problem 6

Consider an "all-pay auction" with two players (the bidders). Player 1's valuation \(v_{1}\) for the object being auctioned is uniformly distributed between 0 and 1. That is, for any \(x \in[0,1]\), player 1's valuation is below \(x\) with probability \(x\). Player 2's valuation is also uniformly distributed between 0 and 1, so the game is symmetric. After nature chooses the players' valuations, each player observes his/her own valuation but not that of the other player. Simultaneously and independently, the players submit bids. The player who bids higher wins the object, but both players must pay their bids. That is, if player \(i\) bids \(b_{i}\), then his/her payoff is \(-b_{i}\) if he/she does not win the auction; his/her payoff is \(v_{i}-b_{i}\) if he/she wins the auction. Calculate the Bayesian Nash equilibrium strategies (bidding functions). (Hint: The equilibrium bidding function for player \(i\) is of the form \(b_{i}\left(v_{i}\right)=k v_{i}^{2}\) for some number \(k\).)

Short Answer

Expert verified
The Bayesian Nash equilibrium bidding function is \(b_i(v_i) = v_i^2\) for both players.

Step by step solution

01

Understanding Uniform Distribution and Player Valuations

Player 1's valuation, \(v_1\), is uniformly distributed between 0 and 1. This means the probability of \(v_1\) being less than any value \(x\) is simply \(x\). The same holds for Player 2.
02

Define Players' Payoffs for Winning and Losing

Let Player \(i\) bid \(b_i\). If Player \(i\) wins (bids higher), the payoff is \(v_i - b_i\); if Player \(i\) loses, the payoff is \(-b_i\). In this all-pay auction, every player pays their bid regardless of winning.
03

Analyzing Symmetry and Identifying Best Response Functions

The game is symmetric, meaning both players face the same decision scenario. Both have valuations following the same distribution. We start by assuming the equilibrium strategy for both players is \(b_i(v_i) = kv_i^2\) and need to determine \(k\).
04

Expected Payoff Calculation for Player 1

Player 1 bids \(b_1 = kv_1^2\). For Player 1 to win, \(b_1 > b_2 = kv_2^2\), which simplifies to \(v_1 > v_2\). The probability Player 1 wins is therefore \(v_1\). The expected payoff becomes \(v_1(v_1 - kv_1^2)\) since \(v_2\) is drawn from a uniform distribution.
05

Maximizing Expected Payoff with Respect to k

To find optimal \(k\), maximize Player 1's expected payoff. The payoff is \(v_1^2 - kv_1^3\), so taking the derivative with respect to \(k\) gives \(-v_1^3\). Setting this to zero indicates the equilibrium condition.
06

Solve for k in Equilibrium

From the derivatives and expected payoffs, set \(-kv_1^3 = 0\). Solving reveals \(k = 1\). Thus, the bidding function is \(b_i(v_i) = v_i^2\).
07

Confirm Symmetry and Equilibrium

Confirm similar calculations for Player 2 provide the same \(b_i(v_i) = v_i^2\), verifying symmetry and Nash Equilibrium conditions in strategies for both players.

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Key Concepts

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

All-Pay Auction
An all-pay auction is a unique type of auction where every participant must pay their bid amount, irrespective of whether they win or lose. This can be thought of as both players "paying the price" for merely being in the competition. Let's break down exactly what this means:

- **Payoff Structure**: For the winner of the auction, the payoff is the difference between their valuation of the object and their bid amount. If a player loses, their payoff is negative, equal to the amount they bid. This reflects the cost they incurred by participating but not winning. - **Strategic Implications**: Due to this payment structure, players must carefully consider how much they are willing to pay for the chance to win, as they must balance between the likelihood of winning and the cost of their bid. Players need to strategize their participation intensely since overbidding can lead to losses even as a winner.

Understanding all-pay auctions allows us to delve deeper into the players' decision-making processes and their optimal strategies.
Uniform Distribution
The concept of uniform distribution in this context implies that each player's valuation of the auctioned object is equally likely to be any value between 0 and 1. Each player knows their valuation but not their opponent's.

- **Probability Characteristics**: The probability that a player's valuation is less than any given number, say x, is exactly x. This straightforward distribution makes the auction calculations easier, as every potential value for a player is equally probable. - **Symmetry and Predictability**: Due to this uniform spread, players can only make educated guesses about the opponent's valuation. They know their own valuation is one realization of this uniform distribution, and they assume so is their opponent's. This lack of skewness allows players to form straightforward predictions, essential for calculating strategic moves.

Grasping uniform distributions is crucial to understanding the equitable conditions under which players estimate their bids in the all-pay auction.
Game Symmetry
Game symmetry refers to the situation where players in the auction have identical strategic conditions. Here, this symmetry stems from three main factors:

- **Identical Valuation Distributions**: Because both players have valuations drawn from the same uniform distribution, they are on equal footing regarding the possible values of the auctioned item. - **Mutual Ignorance**: Each player knows only their own valuation. Since neither knows the other's valuation, they must base their strategies on the same set of assumptions. - **Strategic Equilibrium**: With symmetry, both players end up using the same bidding strategy at equilibrium. It鈥檚 a component of a fair game, where neither player has a probabilistic advantage purely from the structure of the game.

Understanding game symmetry is essential to appreciate why both players adopt identical bidding strategies in the Nash Equilibrium.
Bidding Strategies
Developing effective bidding strategies in an all-pay auction requires analyzing the expected payoffs and understanding the equilibrium conditions. For our players:

- **Equilibrium Strategy**: The equilibrium bidding function given by the solution is of the form, where each player bids in proportion to the square of their valuation, specifically, using a bid function: \[ b_i(v_i) = k v_i^2 \] After analysis, it turns out that the optimal strategy is simply \( b_i(v_i) = v_i^2 \), meaning the constant k is equal to 1.- **Expected Payoffs**: Each player maximizes their expected utility by balancing the bid's cost with the benefit of winning. They use predictions about the other's bid, given that both are reasoning the same way.- **Strategic Analysis**: By setting up their bids using this quadratic relationship, players ensure they aren't overpaying relative to their valuation and likely win chances, a key to making sound strategic decisions.

Formulating bidding strategies involves mathematical reasoning to ensure minimal losses and possible gains from the auction, embodying the core logic of the Bayes-Nash Equilibrium.

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Most popular questions from this chapter

Suppose that you and two other people are competing in a third-price, sealed- bid auction. In this auction, players simultaneously and independently submit bids. The highest bidder wins the object but only has to pay the bid of the third-highest bidder. Sure, this is a silly type of auction, but it makes for a nice exercise. Suppose that your value of the object is 20 . You do not know the values of the other two bidders. Demonstrate that, in contrast with a second-price auction, it may be strictly optimal for you to bid 25 instead of 20 . Show this by finding a belief about the other players' bids under which 25 is a best-response action, yet 20 is not a best-response action.

Complete the analysis of the second-price auction by showing that bidding one's valuation \(v_{i}\) is weakly preferred to bidding any \(x

Consider the common-value auction described in this chapter, where \(Y=y_{1}+y_{2}\) and \(y_{1}\) and \(y_{2}\) are both uniformly distributed on \([0,10]\). Player \(i\) privately observes \(y_{i}\). Players simultaneously submit bids and the one who bids higher wins. (a) Suppose player 2 uses the bidding strategy \(b_{2}\left(y_{2}\right)=3+y_{2}\). What is player 1's best-response bidding strategy? (b) Suppose each player will not select a strategy that would surely give him/her a negative payoff conditional on winning the auction. Suppose also that this fact is common knowledge between the players. What can you conclude about how high the players are willing to bid? (c) Show that for every fixed number \(z>0\), there is no Bayesian Nash equilibrium in which the players both use the bidding strategy \(b_{i}=y_{i}+z\).

Consider the two-bidder auction environment discussed in this chapter, where the bidders' values are independently drawn and distributed uniformly on the interval \([0,900]\). Demonstrate that, by setting a reserve price, the auctioneer can obtain an expected revenue that exceeds what he would expect from the standard first- and second-price auctions. You do not have to compute the optimal reserve price or the actual equilibrium strategies to answer this question.

Suppose you and one other bidder are competing in a private-value auction. The auction format is sealed bid, first price. Let \(v\) and \(b\) denote your valuation and bid, respectively, and let \(\hat{v}\) and \(\hat{b}\) denote the valuation and bid of your opponent. Your payoff is \(v-b\) if it is the case that \(b \geq \hat{b}\). Your payoff is 0 otherwise. Although you do not observe \(\hat{v}\), you know that \(\hat{v}\) is uniformly distributed over the interval between 0 and 1 . That is, \(v^{\prime}\) is the probability that \(\hat{v}
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