Good morning,
I have this game theory problem.
Let's consider 5 farmers, each of them has 2 cows to put into the field. So every farmers can put 0,1 or 2 cows. I denote the three stategies by $q_i$, i=0,1,2. Now, the payoffs ( i.e. the amout of food that each cow will eat) are given by: $$ u_{i}(q_1,...,q_5)=q_{i}(12-(q_{1}+...+q_{5})) $$ Now, for finding the Nash equilibrium I draw the matrix which has on the row one player and on the columns the others four. I observe that strategy 2 stricly dominates 1 so the N.E. is given by (2,2,2,2,2) and the correspinding payoffs are (4,4,4,4,4). (If I estend the strategies to $q_{i}\in[0,2]$ the N.E. remains the same, right?)
Now, the Pareto optimality, if I have understood well should be the strategies (1,1,1,1,1) which give me as payoffs (7,7,7,7,7). Are there other pareto optimality profile? Because I think that I have understood the problem for 2 players but for more I'm not totally sure.
Then, the last question :) If I tax (c) farmers that put two cows in the field, how much I should tax to inforce N.E? For replying to this I have think in that sense: for strategy I have that my payoffs functions are given by $$ u_{i}(q_1,...,q_5)=q_{i}(12-(q_{1}+...+q_{5})-c) $$ Hence, if I look to the payoff matrix in order to have strictly dominance of strategy 2 I should impose c=1.
I hope that someone could help me and check if my idea are corrects.
Thank you in advance!
I have trouble visualizing a 5-d matrix, so prefer a little algebra. Payoff is $q_i(12-q_i-Q_4)$, where I let $Q_4$ be the sum of the other four farmers. To maximize farmer i's payoff, taken other farmers choices as given (the definition of Nash Equilibrium), you get $12-2q_i-Q_4=0$. Assume a symmetric equilibrium (probably unique, but offhand I don't see a quick proof), so that becomes $q=6-4q/2$ or $q=2$, as you discovered by inspection.
Then you correctly point out the NCE is not Pareto Optimal, because you can make everyone better off, if they each only send out one cow. Note the Nash Equilibrium is by definition self enforcing, so I think in part (c) you meant what tax is needed to reach the Pareto optimum.
Now for the tax, do the same analysis but add a tax: $u_i=q_i(12-q_i-Q_4)-tq_i$. Solve for the $t$ that makes $q_i = 1$.