You are playing a simplified version of poker with a friend. The game only has 3

Question

You are playing a simplified version of poker with a friend. The game only has 3 cards in play at any time: a 1, 2, and 3. Larger numbers win. First, two cards are randomly chosen from the deck of 3, one is given to you and one is given to your friend. You go first. Your friend is forced to put 1 dollar in the pot. Now, you can either fold or raise. If you fold, your friend wins. If you raise, you must put 2 dollars in the pot. Now your friend can either call or fold. If your friend folds now, you win. Otherwise, the person with the larger number wins. If both players play optimally, what is your expected winnings from the game?

Explanation / Answer

So here there will be 6 cases

case 1: you have 3 and your friend has 2 : Friend is forced to put 1$ once you raised the bid your friend will fold because the assumption is both the players play optimally so if your friend has 2 and you raise that means your friend know that you have 3. So in this case your gain is 1$

Case2: you have 2 and your friend has 1: In this case you will gain 1$ since your friend will never raise the bid because he has the lowest number

Case 3: you have 3 and your friend has 1: In this case you will gain 1$ since your friend will never raise the bid because he has the lowest number

Case 4: you have 1 and your friend has 3: You loss will be 1$ as you went 1st and you will never raise

Case 5: you have 1 and your friend has 2: You loss will be 1$ as you went 1st and you will never raise

Case 6: you have 2 and your friend has 3: In this case both of you have put 1$ now it's your turn and you don't know wheter your friend has 1 or 3 so in this case both 1 and 3 carries 0.5 probability to be held by your friend and so if your friend has 1 you will win 1$ by raising and if your friend has 3 you will loose 2$ if you raise and if you fold in both the cases you will losse 1$ so this comes to the probablity as (0.5*0.5*1(1st case of raise)-0.5*0.5*2(2nd case of raise)-0.5*1(both cases of fold))= so the gain=0.25-0.5-0.5=-0.75 or loss of 075$

So expected gain=1+1+1-1-1-0.75=0.25$ since this is your expected gain just because you went 1st.

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