Compute a deterministic strategy to sustain the highest feasible discounted payoff in the battle of the sexes

feasible.c

#include <stdio.h>
#include <gmp.h>
#include <stdlib.h>

int main(int argc, char *argv[]) {
    if (argc <= 1) {
        printf("Give number of iterations as arguments!\n");
    }

    mpq_t husband, wife, value, delta;
    mpq_init(husband);
    mpq_init(wife);
    mpq_init(value);

    mpq_init(delta);
    mpq_set_ui(delta, 9, 10);

    for (int j = 1; j < argc; j++) {
        if (j != 1) {
            printf("\n");
        }

        int limit = atoi(argv[j]);

        mpq_set_ui(husband, 0, 1);
        mpq_set_ui(wife, 0, 1);

        mpq_set_ui(value, 1, 10);

        for (int i = 0; i < limit; i++) {
            if (mpq_cmp(husband, wife) <= 0) {
                printf("■");
                mpq_add(husband, husband, value);
            } else {
                printf("□");
                mpq_add(wife, wife, value);
            }

            mpq_mul(value, value, delta);
        }

        printf("\n");

        gmp_printf("Husband: %Qu\n", husband);
        gmp_printf("   Wife: %Qu\n", wife);
    }

    return 0;
}

feasible.py

#!/usr/bin/python

# Slightly more flexible (but less precise for large numbers)
# Python implementation.
# Allows experimenting with periodic strategies.

from __future__ import print_function

d = 9.0 / 10.0

def partial_sum(d, n):
    return sum(d**k for k in range(0, n + 1))

def infinite_sum(d):
    return 1 / (1 - d)

def value(d, start, factor, offset):
    return infinite_sum(d**factor) * d**(start + offset)

def payoffs(d, start, pattern):
    payoff_1 = 0
    payoff_2 = 0

    for i, letter in enumerate(start):
        if letter == "1":
            payoff_1 += 2 * d**i
            payoff_2 += 1 * d**i
        elif letter == "2":
            payoff_1 += 1 * d**i
            payoff_2 += 2 * d**i

    for i, letter in enumerate(pattern):
        if letter == "1":
            payoff_1 += 2 * value(d, len(start), len(pattern), i)
            payoff_2 += 1 * value(d, len(start), len(pattern), i)
        elif letter == "2":
            payoff_1 += 1 * value(d, len(start), len(pattern), i)
            payoff_2 += 2 * value(d, len(start), len(pattern), i)

    return (1 - d) * payoff_1, (1 - d) * payoff_2

s = ""

husband, wife = 0.0, 0.0

for i in range(0, 100):
    print(s.replace("1", "_").replace("2", "X"), (husband, wife))

    husband, wife = payoffs(d, s, "")

    if husband <= wife:
        s += "1"
    else:
        s += "2"

■□□■□■■□□■
Husband: 3238351489/10000000000
Wife: 327486411/1000000000

■□□■□■■□□■■□■□□■□■■□□■■□■□□■■□□■□■■□■□□■□■■□■□□■□■
Husband: 49739528258046340800842059722946801782876073783789/100000000000000000000000000000000000000000000000000
Wife: 4974509422122164786812147914728757694442181869421/10000000000000000000000000000000000000000000000000

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Husband: 499986726565125620045726400063973312900805405218007804755875999039157324275124003971303229076981011/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Wife: 4999867120359867924773397218038231091365116714367268557945265434646687664757858658103973324531145889/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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Husband: 4999999999999999999999933742136868229403459326989431080103889925348554177623688440037587758425104273994244690259045809553870722379066801984174059430640922079574664279659657851168570226305217997938923283515081979443302963469496914369138408064731770019521633554561709687387774978630358910368095577041144455982458437236469589542352441097389915806097771721649974374100923343753339973135822887840990434781458918964392904197845933799446182507774226635536492426248817205263788483132615147241422306836978011/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Wife: 49999999999999999999999340507811836899328516274845937054595135324481706075595450679255895587151610555158012319095691296399328866431635107575903454739008678585341491860678163534640270116722621699806889150074951240725695974904913267690574790404866676710345602888209849039447759707518285415974638682534158401279598162267049188440254758757537063356732054367101948371093848006062515369704161748347923805825472114839305939080952552945112085250818869541820725351863080781530104554307716354483008028774999889/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000