Question

In the years 2000 and 2001 , the bubble burst for many Internet and computer firms. As they closed shop, some of the firms had to liquidate sizable assets, such as inventories of products. Suppose eToys is going out of business and the company seeks a buyer for a truckload of Elmo dolls in its warehouse. Imagine that eToys holds an auction on eBay to sell the dolls and that two retailers (players 1 and 2 ) will bid for them. The rules of the auction are as follows: the retailers simultaneously and independently submit sealed bids and then eToys gives the merchandise to the highest bidder, who must pay his bid. It is common knowledge that the retailer who obtains the load of dolls can resell the load for a total of \(\$ 15,000\). Thus, if player \(i\) wins the auction with bid \(b_{i}\), then player \(i\) 's payoff is \(\$ 15,000-b_{i}\). The losing retailer gets a payoff of \(\$ 0\). If the retailers make the same bids \(\left(b_{1}=b_{2}\right)\), then eToys declares each player the winner with probability \(1 / 2\), in which case player \(i\) obtains an expected payoff of \((1 / 2)\left(15,000-b_{i}\right)\). What will be the winning bid in the Nash equilibrium of this auction game? If you can, describe the equilibrium strategies and briefly explain why this is an equilibrium. (Hint: This is similar to the Bertrand game.)

Short answer

Both bid $15,000, ensuring zero profit margin, in a Bertrand-like equilibrium.

Step-by-step solution

Understand the Auction Setup

In this auction setup, retailers (players) 1 and 2 independently submit sealed bids to obtain a truckload of Elmo dolls, resalable for $15,000. The winner of the auction is the highest bidder who pays their bid. If bids are equal, each has a 50% chance of winning and pays their bid.

Define the Payoff Function

If retailer \(i\) wins with a bid \(b_i\), their payoff is \(15,000 - b_i\). If they lose, the payoff is \(0\). If both bid the same amount, \(b_1 = b_2\), each expects a payoff of \(\frac{1}{2}(15,000 - b_i)\).

Set Up the Nash Equilibrium Conditions

A Nash Equilibrium in this context means that neither player can improve their payoff by changing their bid, given the bid of the other player. Each player's strategy should be their best response given the expected actions of the other.

Consider Possible Bidding Strategies

Retailers will avoid bidding their maximum value of $15,000 since any higher bid would result in zero profit. They may also consider bidding slightly less than the competitor to secure the win.

Analyze Bidding Below Maximum Value

Bidding even a penny below $15,000 gives a positive payoff (e.g., $15,000 - 14,999 = 1$) and ensures winning unless the other player also bids $14,999. However, this line of reasoning suggests that bids could potentially go lower.

Consider the Equilibrium Bid Resembling Bertrand Game

In a Bertrand-like equilibrium, both retailers have an incentive to marginally undercut each other until profit margins reduce to zero. Hence, the Nash Equilibrium occurs when both players bid $15,000.

Key concepts

Nash Equilibrium

The concept of Nash Equilibrium is fundamental in game theory, where it represents a situation in which no player can benefit by changing their strategy while the other players keep theirs unchanged. In the context of the auction described, finding the Nash Equilibrium involves determining the optimal bid for each player, such that neither player can increase their payoff by changing their bid given the opponent's bid.

In this auction, players 1 and 2 are aware that a bid of $15,000 or higher yields zero profit, as it's equal to or more than the resale value of the dolls. Therefore, they need to find the optimal bid below $15,000 that cannot be improved upon without making a loss or getting zero payoff. This dynamic often leads to the bids gradually decreasing until a stable point — the equilibrium — is reached, where both players are satisfied that any deviation from their current bid would worsen their outcome.

Auction Theory

Auction theory examines the strategic behavior of participants in an auction setting, exploring how different auction designs and bid strategies can affect the outcome. In this particular setup on eBay, known as a sealed-bid auction, each player submits a bid without seeing the other’s bid. Then, the highest bidder wins the auction and pays the amount they bid.

Key components relevant to this auction include:

• The sealed bid process, which requires strategic thinking because each player's decision is made under uncertainty about the other's bid.
• The common knowledge of the resale value of $15,000, which provides a ceiling for logical bids.
• The possibility of a tie, resolved by an equal probability chance, adding a layer of risk and expected value calculation.

Auction theory thus helps frame the players' understanding of maximizing their payoff through strategic bidding considering these factors.

Payoff Function

The payoff function is a crucial tool in game theory to quantify the profit or loss a player experiences from a particular strategy or set of actions. In this auction scenario, the payoff for a retailer is defined based on whether they win or lose the auction and by how much they outbid the competitor.

• If a player wins with a bid of \(b_i\), their payoff is the difference between the resale value and their bid: \(15,000 - b_i\).
• If the player's bid loses, their payoff is \(0\) as they neither pay nor receive anything.
• In the event of both players submitting the identical bid, each has an expected payoff of half the difference, expressed as \(\frac{1}{2}(15,000 - b_i)\).

Understanding and calculating this payoff allows players to decide how much risk to take when placing their bids.

Bertrand Competition

In economics, Bertrand competition refers to a market model where firms determine prices rather than quantities. This theory is relevant to the auction problem because it showcases a strategic interaction where participants try to undercut each other until they reach a point of zero economic profit.

In the given auction, this means both retailers will keep lowering their bids to just undercut their competition, leading to a scenario where both could bid down to the marginal cost, which in this case is zero profit with a pitch-perfect strategic play like in Bertrand.

• The competition parallels this auction's setup, as each retailer aims to narrow their bid enough to win, yet not so low that they make a loss.
• This results in increasing pressure for the bids to align closely to the theoretical value of $15,000.

Thus, the Nash Equilibrium here results in both players submitting high bids around their valuation ceiling or in a unique existence where both bid the full possible value, just as predicted by the Bertrand model.