Exercises

1. Complete a game tree analysis for an auction in which the stake is 3 units and the maximum amount a player can bid is 4 units.
2. Suppose our players no longer follow the conservative convention for equivalent bids but decide that if two or more bids lead to the same outcome they will choose the bid which leads to the result least preferred by the opponent. If they lead to equivalent outcomes for the opponent then they will use the conservative convention. This is called the punishing convention. Conduct a game tree analysis of an auction in which the stake is 3 units and the maximum amount a player can bid is 4 units and the players follow the punishing convention. Would you choose to play with the conservative convention or the punishing convention?
3. Suppose the auction rules are changed so that the second highest bidder only pays half of what he or she bid (but conservative bidding still holds). Do a game tree analysis of an auction which uses these rules and in which the stake is 3 units and the maximum amount a player can bid is 4 units.
4. Use O'Neill's theorem to determine the optimal opening bid in a dollar auction if the units of bidding are dimes and the maximum amount a player can bid is $5.