Question

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.

Short answer

Bidding 25 can be optimal if opponents' bids are guessed between 15 and 24. Bidding 20 may miss value opportunities.

Step-by-step solution

Understanding the Problem

In a third-price auction, the winner (highest bidder) pays the amount of the third-highest bid. You value the item at 20, but consider if bidding 25 might be better under certain conditions.

Define the Strategy and Opponent Behavior

We need to consider a scenario where bidding 25 helps us win and pay less than the item's value. Suppose you believe each of your two opponents will bid uniformly between 15 and 24.

Outcome of Bidding 25

If you bid 25 and your opponents bid between 15 and 24, you win the object. The critical condition is when one opponent bids 24 (second-highest) and the other bids 23 (third-highest). You pay 23, which is less than your value 20.

Outcome of Bidding 20

If you bid 20, consider a situation where one opponent bids at 24 and the other at 19. You lose, as the highest bid overtakes yours, and you pay nothing but obtain no value either.

Comparing Expected Values

By bidding 25, you guarantee winning the object when prices reach this critical interval (23 and 24 bid by opponents), paying less than your value and thus gaining utility. Bidding 20 risk losing in similar circumstances where you could have won with a higher bid.

Key concepts

Third-Price Auction

In a third-price auction, three bidders participate and each simultaneously submits a secret bid for an item. The bidder with the highest bid wins the auction. However, what makes this auction unique is that the final price paid by the winner is the amount of the third-highest bid, not their own bid. This intriguing aspect of third-price auctions creates strategic complexities as bidders try to navigate their bids. Although this format is unusual and not common in practice, it illustrates key game theory principles and helps deepen understanding of strategic decision-making.

Sealed-Bid Auction

A sealed-bid auction is a type of auction where each bidder submits their bid in secret, without knowing the bids of others. This contrasts with an open auction, like an English auction, where bids are made publicly and incrementally. In sealed-bid auctions, strategic decision-making plays a crucial role. Because no bidder can predict others' offers, each must carefully consider the value of the item, their own valuation, and possible valuations of competitors.
This consideration must align with how much they are willing to pay — not too little to avoid losing out and not too much to avoid overpaying. Thus, sealed-bid auctions are an excellent example of how uncertainty can affect strategy.

Best-Response Action

A best-response action in game theory is the strategy that yields the highest payoff for a player, given the strategies chosen by other players. It is essentially an optimal reaction to the perceived choices of others in the game. To determine the best-response action in a third-price, sealed-bid auction, a player must forecast others' bids and select a bid amount that either maximizes gains or minimizes potential losses. For example, if a player believes opponents will bid between 15 and 24, a bid of 25 might ensure winning while still paying below one's own valuation of the item. This approach confronts the traditional assumption from second-price auctions where bidding one's true valuation is generally optimal.

Strategy Formulation

Strategy formulation involves deciding upon the plan of action that aligns with the predicted behaviors of other players. In the context of a third-price, sealed-bid auction, formulating a strategy requires analyzing the possible outcomes based on estimated opponent bids.
Being successful in such an auction requires flexibility and insight into potential bid distributions of opponents. A player with a value of 20 for the auction item might consider bidding higher if they believe competitors will bid in a specific range that allows for strategic gain, such as bidding 25 to win the object by paying less than 20. Formulating this strategy draws from basic principles of game theory, such as anticipation, adaptation, and prediction of others' actions. Ultimately, the goal is to adjust one's own bidding to achieve the highest possible utility.