Amy and Becky are in an auction for a painting. The rules of the auction are:
- Bidders must submit one secret bid.
- The minimum amount to bid is $10 and can be done in increments of $10
- Whoever bids the highest wins and will be charged an amount equal to the bid amount.
- In the event of a tie, the auction house will choose a winner based on a lot with equal probabilities.
Amy thinks the fair value of the painting is $56, and Becky thinks it’s $52. These valuations are also known to both. What should be the optimal bid?
The payoff matrix is:
| Becky | |||
| Bid $40 | Bid $50 | ||
| Amy | Bid $40 | 8, 6 | 0, 2 |
| Bid $50 | 6, 0 | 3, 1 |
If Amy bids $40 and Becky $50, Becky wins the painting and her payoff is the value she attributes (52) – the payment (50)= $2. The loser wins or loses nothing.
If Amy bids $50 and Becky $40, Amy gets it, and her payoff will be the value (56) – what she pays (50) = $6.
If both bid at $40, Amy can get a net gain of 16 (56 – 40) at a probability of 0.5. That implies the payoff for Amy = $8. For Becky, it’s $12 x 0.5 = $6.
If both bid for $50, Amy’s payoff = $6 x 0.5 = $3 and Becky’s = $2 x 0.5 = $1
At first glance, it seems obvious that the equilibrium is where both are bidding for $40. If Amy thinks Becky will bid $40, Amy’s best response is to bid the same as her payoff is $8. The same is the case for Becky.
But if Amy expects Becky to go for $50, then Amy must also bid $50. Becky will also go along the same logic.
