So I have this problem from Game Theory and I have most of the solution, but at the end it involves an algebraic inequality that seems un-solvable or at least not solvable within reason.
So the problem goes, two birds are at a beach, Irene and Jonathan, and they love shellfish. But to get at the food, they have to carry it up and drop from on high. To carry it up incurs a cost $c>0$ but getting the food incurs a gain $v>c$. However, birds will often wait for another to catch and drop food, and eat the food themselves without incurring the cost. Thus on any date, Irene and Jon play the following game.
Up Down
Up -c,-c -c, v
Down v,-c 0,0
They play this game infinitely on dates $t=0,1,...$. Find the interval of discount factors $\delta\in (0,1)$ such that the following strategy is a subgame perfect equilibrium: Irene goes Up on Sundays when Jonathan goes Down, starting the first day. Thus on dates 0, 7, 14, ... Irene goes Up and Jonathan Down. Irene goes Down and Jon Up on every other day. If anybody deviates they both stay Down thereafter.
I reason that, if anyone has incentive to deviate, it's Jonathan, so let me investigate that first. If Jon deviates, then on the first day he still stays down and gets $v$ but thereafter gets 0, so that the payoff for cooperating must beat $v$.
So now I need to calculate the payoff for cooperating which I calculate below.
$$v-\delta c-\delta^{2}c - ... - \delta^{6}c$$ $$+ \delta^{7}v - \delta^{8}c - ... - \delta^{13}c $$ $$...$$
is equal to
$$v(1+\delta^{7}+\delta^{14} + ...)-c( [1+\delta+\delta^{2}+...]-[1+\delta^{7}+\delta^{14}+...])$$
is
$$v\left(\frac{1}{1-\delta^{7}}\right)-c\left(\frac{1}{1-\delta}-\frac{1}{1-\delta^{7}}\right)$$
Now I have to set this $\geq v$ and solve for all such $\delta$ but this seems to entail solving an eighth degree polynomial that is not of any nice form that I recognize. Did I screw up, is there some solution method I'm not seeing, or is this just a crazy thing to have to solve for an intro Game Theory question?