Question

Provide a mathematical description of an ascending auction. As an ascending auction is a game, you must specify players, actions, information, strategies, and payoffs.

To help guide your thinking, think about how an ascending auction works. There are discrete rounds t ∈ {1, 2, . . .}. The price at round t is denoted by pt. If the price rises by a fixed ε > 0 each period, what is the price at pt?

Next, think about what the the decision of a bidder is at each time t.

How would you represent this?

What is the information available to a bidder at time t? How can you encode this information?

Finally, how are payoffs determined?

Answer 1

An ascending auction is a game where bidders incrementally increase their offers over discrete rounds. The price at each round t is pt = pt-1 + ε, starting at a fixed price p1. Bidders use strategies based on available information, and payoffs are determined by the surplus value gained over the final price paid.

Step-by-step explanation:

Mathematical Description of an Ascending Auction

In an ascending auction, the game involves a set of players (the bidders) and typically one seller. Each player bids on the item by stating a price they are willing to pay. The auction proceeds in discrete rounds, denoted as t ∈ {1, 2, …}. The price at round t is denoted by pt. If the price rises by a fixed ε > 0 each period, then the price at round t is pt = pt-1 + ε, with p1 being the starting price.

The decision of a bidder at each time t is whether to stay in the auction by bidding the current price pt or to drop out. The information available to a bidder at time t includes all previous bids and the current price pt. This information could be considered as either perfect or imperfect, depending on whether all bidders have the same knowledge or not. In the case of asymmetric information, some bidders might have more information about the true value of the item than others, which can significantly affect their strategies.

The strategies a bidder could use include bidding aggressively early on to signal high value, or bidding just enough to stay in the game and outlast other bidders. The payoffs are determined by the final price paid for the item if the bidder wins, relative to the bidder's valuation of the item. If a bidder values the item more than the final price paid, they gain a surplus. If they value it less, they incur a loss. Therefore, bidders aim to maximise their surplus.

Ultimatum Game's Influence on Auction Strategies

In the context of the ultimatum game, we observe how the decision-making process and strategic behavior in economic games can affect auction strategies. Similar to how Player A decides on an offer to Player B, bidders must consider how much to bid above the current price pt to balance their chance of winning against their potential payoff.