Game Theory - Contributing to a public good

Question

I have attempted to answer the question but I think I am trying to answer it in a very difficult way as the algebra gets messy and confusing.

If anyone could help me out it would be greatly appreciated.

Thanks in advance

Answer 1

You need use the fact that at equilibrium, neither player can improve her payoff by changing her contribution, so the derivative of the payoff with respect to the contribution must be $0$:

$$ \frac{\partial}{\partial c_i}\left(w+c_j+(w-c_i)(c_i+c_j)\right)=-(c_i+c_j)+(w-c_i)=w-c_j-2c_i=0\;. $$

Subtracting these two equations yields $c_1=c_2=c$, as expected, and then either of the equations becomes $w=3c$, so $c=w/3$.

The corresponding payoff is

$$ w+\frac w3+\left(w-\frac w3\right)\left(\frac w3+\frac w3\right)=\frac43w+\frac49w^2\;. $$

The payoff for $c=w/2$ is

$$ w+\frac w2+\left(w-\frac w2\right)\left(\frac w2+\frac w2\right)=\frac32w+\frac12w^2\;, $$

so, as claimed, independent of $w$ that payoff is higher than the equilibrium payoff.