# Optimal strategy for this d6 game

In my attempts to learn the foundations, I was given the following game in a mock interview:

Two players each roll a d6, and are not able to see each other's rolls. The player with the higher value wins \$1. After the players roll their dice, each player may either pay \$0.25 to increase their individual roll by 2 or keep their roll. What is the optimal strategy and payout? Consider the cases for which i) nothing happens when both players roll the same number, and ii) the players keep rerolling (for free) until different numbers appear.

What is the best strategy here? Is it important for both players to have the same strategy for Nash/symmetric equilibrium (not sure what the terminology I should be using is)? What should my considerations be in calculating the expected value? Please help prod me to attempt deriving the strategy, thanks!

## 1 Answer

Let the probability distribution of the other person's roll be $$P(i)$$.

Hint: If you roll a $$n$$, what will make you indifferent to pay $$0.25$$ to increase your roll?

You are indifferent if $$P(n) + P(n+1) = 0.25$$.

Is there a probability distribution when you are always indifferent?

The probability distribution is $$P(i) = \frac{1}{8}$$. (Each outcome is equally likely)

If yes, can that be achieved via the rules?

It can be achieved. Specifically, what is the strategy?

If yes, is that a symmetric Nash equilibrium?

Yes, see comment.

If yes, what is the payout?

The payout under case 1 is $$\frac{5}{16}$$, under case 2 is $$\frac{1}{2}$$. Why?

Is this the max possible payout? Why, or why not?

Case 1: (I had an error so now I'm not certain)

Case 2a: My interpretation for Case 2 was that they don't have to pay the \$0.25 and get to reroll. If so, the total payout in each turn is 1, and since the strategy is symmetric, hence the max payout is $$\frac{1}{2}$$. Case 2b: An equally valid interpretation is that they will have to pay the \$0.25 regardless. Under this interpretation, it's not immediately clear to me what the strategy should be.

• Hi, thanks for the quick answer — isn’t the PMF itself affected by the strategy? For example whether I choose to increase when I roll a 6 will affect the probability mass of attaining an 8? – user107224 Nov 13 '19 at 10:28
• Your opponent's pdf is fixed. Determine your best response to a given distribution. – Calvin Lin Nov 13 '19 at 10:29
• sorry for another silly question — but I don’t see why the opponent’s PMF should be fixed. Are they also not trying to optimise their strategy according to what they think our strategy is too? – user107224 Nov 13 '19 at 10:31
• For NE, each strategy is a Best Response to the other. So, a good first step is "Given any strategy (fixed pdf), what are the best response?". After that, identity pairs of mutual best responses, which gives us a NE. In particular, if there is a strategy S that the best response is every strategy (IE you're indifferent), then (S, S) is a NE. As an example, in Rock Paper Scissors. The play-all-equally strategy $S = (1/3,1/3,1/3)$ makes the other player indifferent since the expected win = 0 always. (Note: I'm not saying that this is the only way to obtain a NE, as it is not). – Calvin Lin Nov 13 '19 at 10:36
• Neither am I claiming that "this sequence of steps is how you should approach all such problems". I'm claiming that "this sequence of steps works for this problem". – Calvin Lin Nov 13 '19 at 10:42