Here are two challenge problems based on the AKQ game. Challenge problems are open until the first good solution is accepted, and may be reopened if a later, better solution is found. Winners of challenge problems receive the undying glory and admiration of myself and their classmates, and have an opportunity to present their solutions to the class. Note that the intent of challenge problems is to encourage you to think creatively about “open” problems; it is possible, though, that some of these problems have been considered by others before and you could find answers on the web. This is not encouraged, but if you do find a published solution, you can still submit it (crediting the source, of course), and although it will not cover you with as much glory as an original solution, it may still be worth presenting.

## Strategic Options for Player 2

We determined that the original AKQ game is advantageous to Player 1: when both players follow optimal strategies, player one expects to make a profit. One way to make the game more fair might be to give Player 2 more strategic options. In the “full-street” AKQ game, the rules are:

• Player 1 can bet or check one unit. (as in the original game).
• Player 2 can either (1) fold, (2) call, or (3) bet/raise one unit (adding new strategic option #3).
• If Player 2 bets/raises, Player 1 now has the option to either fold or call.

Describe a solution to this game, explaining the optimal strategies for each player and the expected value for Player 1.

## Unstructured Game

In the original game, all best must be one unit. Consider what happens when this restriction is relaxed, so now Player 1 can bet any amount between 0 (equivalent to checking) and his total stack (n1). Assume the bet has no quantization, and can be any real number between 0 and n1. Then, Player 2 either folds or calls. Note that if the bet exceeds Player 2′s stack, Player 2 can call (“all-in”) and Player 1′s effective bet is the size of Player 2′s stack (n2). As in the original game, each player antes one unit before the action. (What matters, of course, is the ratio between the bets and the antes, not the absolute values.)

How does this impact the optimal strategies for each player and expected value of the game for Player 1?