Alternating Die-Roll Game: Optimal Accept/Reject Strategy

Two players, A and B, take part in a game using a fair 21-sided die; its faces carry the integers from $-10$ to $10$, one integer per face, with every face equally likely.

A goes first. On a turn, the player whose turn it is rolls the die, looks at the resulting value $v$, and then picks between two options:

• Accept. The game ends at once, and the opposing player pays the roller $v$. (When $v$ is negative, the roller instead pays the opponent $|v|$.)
• Reject. No payment is made, and the turn passes to the other player, who rolls and faces the very same accept-or-reject decision.

Play alternates in this way without limit until someone accepts.

What is the optimal strategy for each player, and what is the expected payoff to A when both play optimally?

hint Where to start Once a rejection occurs, the roles merely swap and what remains is identical to the original game. Look for a stationary strategy — and ask whether A's and B's decision rules should really differ at all.

hint Value recursion Let $V$ be the expected payoff, under optimal play, to whichever player is about to roll. Rejecting hands your opponent a position worth $V$ to them (so worth $-V$ to you). Write $V$ as an expectation over the roll of the better of accepting and rejecting, and solve the resulting fixed-point equation.

hint Common pitfall "Accept anything $\ge 0$" feels natural but is not optimal. Compare taking a small guaranteed loss now against the expected cost of handing your opponent the move.

Constraints & Assumptions

• The die is unbiased: each of the 21 integers from $-10$ to $10$ carries probability $1/21$, and rolls do not affect one another.
• Payoffs are zero-sum: the amount the accepting player gains is exactly what the other player loses.
• The number of turns is unbounded, there is no discounting, and rejecting costs nothing.
• Both players maximize expected value and are risk-neutral, and both play optimally, each knowing the other is rational.

Clarifying Questions to Ask

• Does every face have the same probability, and is each roll independent of the ones before it?
• When B accepts, does A pay B the face value — that is, is the game fully symmetric under a swap of roles?
• Is there any cap on the number of rounds, or any per-round cost or time discounting?
• May a player accept a negative value, and does that mean the roller pays out of pocket?
• Are we maximizing expected value (risk-neutral), or should risk aversion factor into the strategy?

What a Strong Answer Covers

Follow-up Questions

• If the roll becomes continuous and uniform over $[-10, 10]$, how do the strategy and the game's value change?
• Suppose rejecting imposes a fee $c > 0$ on the player who rolls. How do the thresholds and the value shift?
• What if all faces are positive — for instance $1$ through $20$? What does that show about where the tension between accepting and rejecting actually originates?
• How could you check your closed-form answer numerically?

Overview: The problem tests probabilistic reasoning, sequential decision-making and zero-sum game analysis, exercising expected-value computation, optimal stopping and reasoning about symmetry.

Community answers

Answer by fineman3301 The optimal rule is to accept any roll of $-1$ or higher. The expected value is $1.8$.