Homework 3

Homework 3 2008 DMA Soccer, Sequential Games with Randomness,

This question refers to DMA soccer discussed here. Assume each team has six players, and A has three defence, two midfield, and one attack players, and Beth has 2 defence, 2 midfield, and 2 attack players.

Analyse the 3·3 version of Polynomial REC THE SQUARE with randomness game. Explain what will happen if both play optimally. How much will each of the players expect, if a win counts a s +1 and a loss as -1?
Note that you can describe the game using the following 27 states, where each symbol subsumes the cases obtained by rotation and reflection.
The analysis is done on this Excel sheet.

The following sequential game with randomness but perfect information is given in extensive form. There are two players, Ann and Beth. Vertices are labeled by the player whose turn is it. Vertices without such a label are random moves, the probabilities for the two options are 50% and 50% in each case. The payoffs of the end vertices are given, always first Ann's payoff, then Beth's.
Here is the solution done by using backwards induction for a "neutral" Beth:

Note that there are a few variants here. The above solution is the case where a player would randomly chose among moves that yield identical payoffs for here (the neutral player). Note that in class we also discussed cases where a player, in such a case where two payoffs of two options for that player are identical, would look at the opponent's payoff as second (minor criterium). A friendly player would try to give the opponent the best possible (all under the condition that for her own payoff it doesn't matter), and a hostile player would try to minimize it. In the game above, Beth is in both situations indifferent about her moves. A friendly Beth would play "1" in both cases, a hostile Beth would play "2" in both cases, a neutral Beth alternates. Whether or not Ann is friendly or neutral or hostile doesn't matter. If Beth is friendly, then the values of the game are 2 for Ann and 1 for Beth. If Beth is neutral, then they are 1.5 for Ann and 1 for Beth, and they are 1 for both in case of a hostile Beth. In all these three cases, Ann would start the game by playing "1".

Consider the following game:

KUHNPOKER: We play with a 8 cards deck, four "1"s and four "2"s. Every player gets one card and looks at it secretly. The start bet is $2. Ann moves first by either checking or raising.

Draw the extensive form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?

Since Ann has 4 information sets with two moves possaible in each, she has 16 pure strategy. The same holds for Beth.