.. _Gory_Details:

The Two Dice Variables
======================

The die variable represents the outcome of rolling a two-sided die. We assume that both Arne and Bente are playing with non-skew dice, so for both variables there is a 50% probability to obtain "1" and 50% for "2".

Arne's Bid Variables
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It is pretty easy to convert our guess of Arne's strategy to conditional probabilities. As an example we show the table of AB1:

.. figure:: images/gory_details.png

The only thing that should be noted is that we in AB2 and AB3 for technical reasons respond to a "call" from Bente with a "call". This is is to ensure that once a game is over no player starts to bid again and no extra utility values are granted for a game that is already won!

The Utility Functions
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This task is the most difficult of the three and you are advised to consult net file for the exact tables. The utility functions can be grouped into two:

1. U5 and U7 which determine the winner of a game.
2. U1U4 and U2U6 which ensure that Bente only performs valid actions and determine the winner of a game.

The first ones are pretty straightforward. In both U5 and U7 it is Arne who's turn it currently is. For each situation where Arne calls the winner of the game is determined from the states of the two die variables. In case Bente wins we assign this situation a utility value of 1, should Bente lose the utility is set to -1.

Situations where Arne doesn't call are so called "neutral situations" since the game continues and no winner is found. Hence we assign neutral situations the value 0.

With our "call->call" trick the situation where both players called corresponds to an already finished game. Bente's outcome has therefore already been found, hence this situation is also assigned the value 0.

For U1U4 and U2U6 some extra work has to be done, because it is Bente's turn. The assigning of values follows the same schema as for U5 and U7: Should Bente decide to call her outcome is determined.

All the other possible bids are either neutral bids since they don't end the game or illegal bids. Neutral bids are assigned the value 0 whereas illegal bids are assigned the value -10 - a real punishment! (the proper value for this situation would be minus infinity, but -10 will do for our game). We enforce our "call->call" trick by making all other bids than call illegal actions once Arne called.