Parity Rock–Paper–Scissors Games and the Alper Constant

Two players choose numbers from \(\{1,\dots,n\}\). Equal numbers draw, an odd difference is won by the smaller number, a nonzero even difference is won by the larger number, and a win with the number \(m\) pays \(2m-h\), where \(0\le h\lt 2\) is fixed. For \(h=1\) this is the game of Problem 1012 of Project Euler; for \(h=0\) the payment is proportional to the winning number.

For every \(h\) outside an explicit countable set \(\mathcal E\) we prove that the game has a unique equilibrium, supported on \(2k+1\) consecutive numbers ending at \(n\) or \(n-1\), and we solve it in closed form. When the support ends at \(n\), the probability \(P(n)\) of the largest number is \(\bigl((2k^2\)\({}+1)R_k\)\({}-2(k^2\)\({}-1)\bigr)\)\(/\bigl(3(2k\)\({}+1)\bigr)\) for an explicit product \(R_k\) that depends only on the largest payment; otherwise \(P(n)=0\). The support widens at thresholds given by the zeros of these functions; for \(h\in\{0,1\}\) the thresholds are cubic polynomials in \(k\) for each parity.

For \(S_h(N)\)\({}=\sum_{n=3}^{N}P(n)\) we prove \(S_h(N)\)\({}=(3\)\(/2)^{4/3}N^{1/3}\)\({}-\tfrac14\log N\)\({}-\mathcal{A}(h)\)\({}-\tfrac32\tau(1\)\({}-\tau)\)\({}+O_h(N^{-1/3})\), where \(\tau\in[0,1)\) is the relative position of \(N\) in its block, and we determine the next term. The constant \(\mathcal{A}(h)\) plays the role that Euler's constant plays for the harmonic series. We call \(\mathcal{A}\)\({}=\mathcal{A}(0)\)\({}=1.7771799879\)\(\ldots\) the Alper constant, give a series and an integral representation, and certify \(80\) decimal places of \(\mathcal{A}\) and of \(\mathcal{A}(1)\)\({}=1.6464176534\)\(\ldots\) with interval arithmetic. No expression of either constant in classical constants was found.

The Alper function \(h\)\({}\mapsto\mathcal{A}(h)\) is strictly increasing and convex between consecutive points of \(\mathcal E\) and jumps down at each of them; it tends to \(\mathcal{A}(1)\) as \(h\to1^-\) and to \(\mathcal{A}(0)\) as \(h\to2^-\), and \(\mathcal{A}(h\)\({}+2)\)\({}=\mathcal{A}(h)\) for \(0\le h\lt 2\), \(h\notin\mathcal E\), when the games are extended to larger parameters.