Two players, A and B, play the following game with a fair 21-sided die whose faces are labeled with the integers $-10, -9, \ldots, 9, 10$ (one integer per face, each face equally likely). Player A moves first. On a player's turn, that player rolls the die, observes the face value $v$, and then chooses one of two actions: - **Accept.** The game ends immediately, and the opposing player pays the roller $v$. (If $v$ is negative, this means the roller pays the opponent $|v|$.) - **Reject.** Nothing is paid, and the turn passes to the other player, who now rolls and faces exactly the same accept/reject choice. Turns alternate this way indefinitely until some player accepts. What is the optimal strategy for each player, and what is the expected payoff to A when both players play optimally? ```hint Where to start After any rejection, the roles simply swap and the remaining game looks **identical** to the original one. 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 fair: each of the 21 integer values $-10$ through $10$ has probability $1/21$, and successive rolls are independent. - The game is zero-sum: whatever the accepting player receives, the other player pays. - There is no bound on the number of turns, no discounting, and no cost to reject. - Both players are risk-neutral expected-value maximizers, and both play optimally (each knows the other is rational). ### Clarifying Questions to Ask - Is every face equally likely, and are successive rolls independent of earlier ones? - When B accepts, does A pay B the face value — i.e., is the game fully symmetric under a role swap? - Is there any limit on the number of rounds, or any per-round cost or time discounting? - Can 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 - Recognizing the stationary, symmetric structure: after any rejection the opponent faces the identical game, so the continuation value is the same at every turn and both players share one decision rule. - Formulating the mover's expected payoff as a fixed-point equation and deriving a threshold (accept/reject) policy from it. - Handling the discrete faces correctly: identifying the integer cutoff, checking the threshold is self-consistent, and computing the exact game value. - Sanity checks: indifference reasoning at the boundary, comparison against the naive threshold-at-zero policy, and an argument that the game terminates with probability 1. ### Follow-up Questions - How do the strategy and the game value change if the roll is continuous, uniform on $[-10, 10]$? - Suppose each rejection costs the rolling player a fee $c > 0$. How does that shift the thresholds and the value? - What happens if every face is positive — say the faces are $1$ through $20$? What does that reveal about where the accept/reject tension actually comes from? - How would you verify your closed-form answer numerically?