The entertaining book The Compleat Strategyst by John Williams contains many simple examples and informative discussion of strategic form games. Here is one of his problems. I know a good game ,\" says Alex. \" We point fingers at each other; either one finger or two fingers. If we match with one finger, you buy me one Daiquiri, If we match with two fingers, you buy me two Daiquiris. If we don\'t match I let you off with a payment of a dime. It\'ll help pass the time. Olaf appears quite unmoved. \"That sounds like a very dull game \" at least in its early stages.\" His eyes glaze on the ceiling for a moment and his lips flutter briefly; he returns to the conversation with: \"Now if you\'d care to pay me 42 cents before each game, as a partial compensation for all those 55-cent drinks I\'ll have to buy you, then I\'d be happy to pass the time with you. Olaf could see that the game was inherently unfair to him so he insisted on a side payment as compensation. Does this side payment make the game fair? What are the optimal strategies and the value of the game?
Now there are 4 possibilities that can happen.
1 . Alex points 1 finger , Olef points 1 finger.
2. Alex points 1 finger , Olef points 2 fingers
3. Alex points 2 fingers , Olef points 1 finger
4. Aex points 2 fingers , Olef points 2 fingers
Now as per the first game plan said by alex , Alex gains 55 cents if event 1 happens and 2*55 =110 cents if event 4 happens. He loses 10 cents if event 3 or 4 happens.
The probability of each event happening is 1/4. So Alex would get an amount = 1/4(55) + 1/4(-10) + 1/4(-10) +1/4 (110) =1/4(145) =36.25 .So he would profitize.
The second strategy gives 42 cents before each game , so the amounts would be;
1/4(-42 + 55) + 1/4(-42 -10) + 1/4(-42-10) +1/4(-42+110) = -42+1/4(145) = -42+36.25 =-5.75. Hence by second strategy , the game would be little unfair to Alex as compared to the first strategy which is higly fair to him.
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