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Two players, 1 and 2, simultaneously choose their consumption of a public good. Given the consumption choices, g1 and g2, player 1 derives a marginal benefit of MB1 = 10 - (g1 + g2), while player 2's marginal benefit is given by MB2 = 8 - (g1 + g2). The unit price of the public good is equal to 4.

The question asks to find a Nash Equilibrium of the game, then calculate the efficient level of public good provision and compare it to the NE.

I can understand the efficient provision of public good which will be nothing but the sum of marginal benefits set equal to the marginal cost. In this case, it turns out to be 7. What will be the Nash equilibrium?

An update with my attempt:

Player 1's best response function: g1 = 6 - g2 Player 2's best response function: g2 = 4 - g1

Added these two to get the total level of public good - it comes to 5.

Then I equated the two response function to get: g1 = g2 - 2

Substituted it in g1 + g2 = 5 equation with getting g2 = 3.5 and g1 = 1.5

I don't know if this is correct.

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For the Nash equilibrium of any simultaneous-play game, you are looking for the point where each player is playing a best response to all other players at the same time.

So your steps to solving this game should be:

  1. Determine player 1's best response function ($g_1$ as a function of $g_2$)
  2. Determine player 2's best response function ($g_2$ as a function of $g_1$)
  3. Now you have two equations with two unknowns. Solve this system of equations algebraically to get the Nash equilibrium $g_1^*$ and $g_2^*$.

So how can you get the best response functions? Take player 1 for example. Player 1 wants to set $MB_1 = MC_1$. You have $MB_1$, and $MC_1 = 4$. So set them equal to each other and solve for $g_1$ to get player 1's best response function $g_1^*$ as a function of $g_2$. Then similarly for player 2 to get $g_2^*$ as a function of $g_1$.

That's not quite all, though. In this game, the players (presumably) can't contribute negative amounts. So the real best response functions will be of the form

$$ g_1^*= \begin{cases} some.function.of.g_2 \text{ if } some.condition.of.g_2\\ 0 \text{ otherwise} \end{cases} $$

So, first, try to solve your two best response functions together algebraically, assuming that both $some.condition.of.g_2$ and $some.condition.of.g_1$ hold so neither is 0. If you don't get a solution, or if that solution makes one of them negative or violates one of the $some.condition$s, then instead try setting $g_1 = 0$, find $g_2^*$, and see if that gives you a Nash equilibrium (i.e. if $g_1^*(g_2^*) = 0$). Then try setting $g_2 = 0$, find $g_1^*$, and see if that gives you a Nash.

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    $\begingroup$ Thank you! I completed the first 3 steps - I can't find a solution. My response function of Player 1 is g1 = 6 - g2. Of player 2: g2 = 4 - g1. If I set g2 = 0, I'm getting g1 as 6 but I don't think that's correct. $\endgroup$ – user708015 Oct 16 at 17:13
  • $\begingroup$ If when you set g_2 = 0, you get g_1 = 6, then you only have one more step to go: if setting g_1 = 6 gets you g_2 = 0 (keep in mind negatives aren't possible), then you have a Nash! $\endgroup$ – NickCHK Oct 17 at 18:11

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