Bidding Strategy (English Auctions)

COMP315 Lectures 2023-02-06

A strategy is a description of how the bidder will behave in every possible situation.

Best Response

Suppose there are $n$ bidders using strategies $s_1,\dots ,s_n$ where you are bidder 1:

• The outcome of the auction when everyone uses that strategy can be computed.
  • We can call the outcome $O(s_1,\dots ,s_n)$
• Your strategy $s_1$ is a best response to $s_2,\dots ,s_n$ if there is no $s_1^{\prime }$ such that $O(s_1^{\prime },\dots ,s_n)$ is better for you than $O(s_1,\dots ,s_n)$.

This model doesn’t account for people who change their strategies.

If $s{j}_1$ is the best response to $s_2,\dots ,s_n$, then other might play different strategies $s_2^{\prime },\dots ,s_n^{\prime }$ if you use $s_1$.

Rationalisability

A strategy $s_1$ is rationalisable if:

1. There are strategies $s_2,\dots ,s_n$ that $s_1$ is best response to.
2. $s_2,\dots ,s_n$ are rationalisable.

Therefore, everyone is playing the best response at least one other person’s strategy, given that everyone is playing a rationalisable strategy.

Not every rationalisable strategy is optimal.

Optimality

A strategy $s_1$ is optimal if it is the best response to every choice of strategies $s_2,\dots ,s_n$ of other bidders:

• If an optimal strategy exists you should use it.
• Otherwise default to a rationalisable one.

Every optimal strategy is rationalisable.

English Auction Strategy

Consider we are in an English auction with the following assumptions:

• There are exactly two bidders.
• The minimum increase is negligible.

• The item is worth $v_1$ to you.

  You should not bid more than $v_1$ otherwise you will be disappointed.

1. Suppose that the current standing bid $b_2$ by the other bidder and $b_2<v_1$.

2. Raise with the following inequality:

   $$b_2<b_1<v_1$$

   where $b_1$ is your new bid.

You can never be certain what your opponent will do. Therefore you must follow these rules as you can’t be sure whether you will be outbid or not.

English Auctions with Minimum Increase

You can make use of the minimum raise to increase your profit:

1. Consider that you know your competitor values the item at £100 and you at £150. There is a minimum raise of £10.
2. If you are able to bit £91 then the other bidder can no longer raise. Therefore you have a profit of £150 - £91 = £59.

Before we would have only gained a profit of £40. By getting the bid right we can increase profit by up to twice the minimum raise.

If your information is false then you can end up making a reduced profit by raising too quickly.