Question

Dollar Auction. (Taylor [36]) The dollar auction is an elementary model of escalatory behavior such as the buildup of American troops in Viet Nam or the arms race of the 1960's, 70's and 80's. It is an excellent vehicle to introduce the mathematical analysis of decision trees. In a dollar auction, two players bid for a STAKE held by an auctioneer. We will use an example in which the stake is $3. Each player has a BANKROLL. In our example each player has a bankroll of $3. The players bid in turn (in whole dollars) until there is a pass. At this point the highest bidder pays her last bid to the auctioneer and receives the stake. The catch is that the other player must pay his last bid to the auctioneer as well. The problem is to determine the best bidding strategy for each player.

Note: To simplify matters, we assume that both players follow the Conservative Convention—If either bidder determines that twd or more different bids will lead to the same eventual outcome for herself, and that no bids will lead to a better outcome, then the bidder will choose the smallest of those bids (that is, the conservative bid). To illustrate

the Conservative Convention, suppose that in our example above, Player 1 bids $2. Now Player 2 can only bid $3. But this will result in a net gain of $0 to Player 2, the same outcome as Passing (bidding $0). Under the Conservative Convention, Player 2 passes, so Player 1 pays the auctioneer $2 and receives the stake of $3, a net gain of $1. Note that if Player 2 had bid $3, then Player 1 would lose $2. This is not allowed under the Conservative Convention. In particular, punishing the other player is not a goal of the
game (although it is a goal in military build-up situations). By the way, we also assume that each player has full knowledge of the other player's strategy, including the fact that the other player will use the Conservative Convention. So I know you will use the Conservative Convention, that you know I know that, that you know I know that you know I know that and so on. And again, making the other player lose money is not a goal.

1. Determine the best bidding strategies if the STAKE is 3 and each player has a BANKROLL 3.

2. Determine the best bidding strategies if the STAKE is 4 and each player has a BANKROLL 3.

3. Experiment with other stakes and bankrolls to see if you can come up with a general rule

You don't need to assume that both players have the same bankroll. Your rule should show that if the stake is 20 and the bankroll is 100 for each player, then the best opening bid is 5

Answer 1

Q) Determine the best bidding strategies if the STAKE is 3 and each player has a BANKROLL 3.

Ans: Consider this as scenario, if stake (s) =3 and bankroll (b) =3, the possibilities are:

if player 1 starts with his bid with $1, the choice with player 2 is go with $2 or $3 or pass,

Player 2 bids:

• Passes, the outcome is $2 for player 1 and 0 for player 2
• Goes for $2, then further if player 1 passes- outcome is $1 for player 2 and -1 for player 1
• Goes for $3 (the maximum bankroll), -outcome is 0 for player 2 and -1 for player 1

In conservative convention, the players try to maximize their profit but doesn’t go for limiting their profit and making the partner go for maximum loss

Therefore, going with this rule, as both players know that everyone goes for conservative convention

If player 1 bids with $1, the player 2 will obviously passes so that outcome will be $2 for player 1 and 0 for player 2

Therefore, the best betting strategies is if player 1 gets a chance, he should directly bid for $2 and get profit of 1 as obviously player 2 will pass

If player 1 bets with $1, player 2 should bid for $2 and come away with profit of $1

2. Determine the best bidding strategies if the STAKE is 4 and each player has a BANKROLL 3.

Ans: Consider this scenario, if stake (s)=4, and bankroll (b)=3, the possibilities are:

If player 1 starts with a bid of $1, then player 2 can go for bid $2, $3 or pass the outcomes are:

• Passes, the outcome is 3 for player 1 and 0 for player 2
• Goes for bid$2, then player 1 either bids for $3 or passes, if player 1 passes, outcomes are -1 for player 1 and 1 for player 2, - it will go as conservative contention because here if player 1 bids, he will have profit. if goes with $3, then outcomes are 1 for player 1 and -2 for player 2, but this is not conservative convention because here player 1 loss is minimized and player 2 loss is maximized
• Goes for bid $3, then player 1 obviously passes and outcome is $1 for player 2 and -1 for player 1- it is conservative convention because here player 2 is getting profit.

Therefore, the best betting strategies is player 1 should go for $3 and get profit 1 because if he bids $2 or $1, player 2 will definitely bids as he can get more profit

For player 2, if player 1 bids 1 he can directly go for bid of $3, so that he can get profit

3. Experiment with other stakes and bankrolls to see if you can come up with a general rule

General theorem: The best betting strategy is player 1 should always subtract the bankroll with s-1 and at the point where b-(s-1) is non negative and less than s, he should start with that bid. If b is less than s, then he should go for s-1 as initial bid as the player 2 would definitely doesn’t bid if he goes with conservative contention.

Similarly, if player 1 bids with any other number apart from b- (s-1), player 2 should immediately bid with either b- (s-1) or s-1 to get his own profit and minimize the loss of player 1.

For example, bankroll (b) =100 and stake (s)=20, the possible bankrolls should be 100-19=81. 81-19=62, 62-19=43, 43-19= 24 and 24-19 would be 5, 5 -19 is -14, so they can’t bid. Therefore, the best bid should be 5

If the bankroll is 100 and stake is 25, the possible initial bid can be 24 or 5 such that he can get a profit of 1 or loss of 5.