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# Two people each bids a number before throwing a 30 faced die. Whoever gets closer to the number wins and wins the amount of money equal to the number they throw. e.g I bid 15 and you bid 16. the die lands on 10 then i win 10 from you. What's the best strategy and the expected payoff.

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We choose 22. If our opponent plays optimally, he chooses 21.Let us see why. Clearly, he is facing a choice between 21 and 23. If he picks 23, his expexted payoff is 8/30 * 53/2 - 22/30 * 23/2 = -41/30. If he goes for 21, his expected payoff is 7/10 * 11 - 3/10 * 26 = -0.1. Note that in both cases our opponent on average loses, so he will choose 21 to minimize his loss. In this case our expected profit is his expected loss, ie on average we expect to make 0.1 per game.

Denys Fridman on

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What was your approach to the problem?

Max on

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I think from question that first you bet, and afterwards your opponent. It's quite obvious that when you bet on one number, for example a, your opponent will bet either on a-1, or a+1, otherwise won't be optimal for him (betting on 1 or 30 isn't very smart). So for every number you bet on, you have two expected gains (depending whether your opponent chooses a-1or a+1). Because your opponent plays optimally, you are looking for number with the highest lower gain of two above-mentioned. If you choose 21, your expected gains are 231/30 and 255/30, if you choose 22 then you will have expected gains 253/30 and 234/30, so therefore choosing 22 is optimal, and your opponent chooses 21.

Anonymous on

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How do you calculate your gain to be 231/30 & 255/30? I'm not sure of how you're arriving at that expected value?

Anonymous on

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For the first case, i.e., your opponent choose a+1, your wining prob. is a/30, and the avg. gain is (1+a)/2, so the exp. gain is a(1+a)/60; for the second case, i.e., you opponent choose a-1, your wining prob. is (30-a+1)/30, and the avg. gain is (a+30)/2, so the exp. gain is (31-a)(30+a)/60. Replacing a with 21, you'll get 231/30 & 255/30.

Jia on

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why 22 vs 21

Anonymous on

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Is the other person's bet known before I have to bet myself?

Anonymous on

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It's just asking what number n is the first number such that the sum of ints from 1 to n is the the smallest sum greater than the (sum of numbers 1-30)/2

Hoonie on

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Answer is 21 VS 22 If you bid X and your opponent bid X+1, your expected gain is 2x^2+3x-899 If you bid X and your opponent bid X-1, your expected gain is -2x^2+2x+930 Given you bid first, solve the equation, you should find that if your bid is below 22, your opponent should bid 1 point higher than you; if your bid is greater and equal to 22, your opponent should bid 1 point lower than you; Now you can calculate X=22 opponent 21 expected gain 6/60 X=21 opponent 22 expected gain 46/60 so the best strategy is you bid 21 and expected gain is 46/60, provided your opponent plays reasonably.

miaoya on

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Sorry the first two lines of expected gain above should divide 60

Anonymous on

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Let n be the sides of the dice. Player 1 needs to pick a number k such that 0

Laksh on

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Let n be the sides of the dice. Player 1 needs to pick a number k such that 0

Laksh on

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Let n be the sides of the dice. Player 1 needs to pick a number k such that 0

Laksh on

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22 vs 21

Anonymous on