Closed-form expression for the attacker’s winning threshold A(t,n,s) in the special-round game

Determine whether a closed-form expression exists for Alice’s winning threshold A(t,n,s) in the game G^1_k, where A(t,n,s) gives the minimal attacker budget (as a multiplicative factor of the defender’s budget) needed to force a win given t total rounds, n defender wins required, and s special rounds; either derive such a closed-form expression or prove that no such closed-form exists.

Background

In the 𝒢^1_k game, some rounds are “special,” requiring the attacker to outbid the defender by a factor k>1 to censor a move. The function A(t,n,s) captures the attacker’s winning threshold, i.e., the smallest attacker budget (relative to the defender’s) sufficient to force a win when there are t total rounds, the defender must win n rounds, and s of the remaining rounds are special.

Theorem 3 (optimal_specials) shows that A is linear in the defender’s budget and depends only on the number of special rounds, with a recurrence and boundary conditions enabling dynamic-programming computation. Despite this structural understanding, the authors explicitly state they were unable to find a closed-form for A and suspect none exists. Establishing the existence or nonexistence of such a closed-form would sharpen the theoretical understanding and simplify practical parameter setting.