<p>The idea behind Pareto optimality is <em>efficiency</em>. On the discrete set, there is no allocation where, compared to $(1,1,1,1,1)$, we could make someone better off and keep the others at least indifferent. This is why this allocation is Pareto optimal. However, any other <em>efficient</em> allocation (in the sense that we are not throwing away resources) will be Pareto optimal here as well. </p>

<p>For the set in $\mathcal R$, we assume a symmetric solution and solve</p>

<p>$$\max_{q\in[0,2]} 5\cdot q(12-(5q))$$
$$\Rightarrow 5(12-5q) -25q$$</p>

<p>The first term is the direct benefit of increasing $q$, the second one is the externality (that we now internalize). The interior solution is given by $q = 1.2$ where everyone gets the payoff $7.2$. We know that the interior solution solves the problem given that the boundaries $0, 2$ both delivered lesser value than $q=1$. </p>

<p>Finally, for the tax, derive the FOC of the agents given a general tax $c$. Then, look for the optimal quantity <em>you want them to have</em>, that is, replace $q_i$ with $q_i^*$, the welfare-maximizing solution, and then look for the value of $c$ that would solve the FOC (and hence make the farmers optimally chose what you want them to).</p>

CC BY-SA 3.0
2015-01-26T19:00:16.013Z
2017-01-03T15:03:07.793Z
2017-01-03T15:03:07.793Z
FooBar
<p>My understanding is that we do not answer homework problems here. (I might be wrong.) However you did lay out some ideas and I will give you feedback on those:</p>

<p>1) There is no guarantuee that the NE will remain the same if you extend the strategy space. You have to check again by looking for the arg max of the payoff function given the strategy profile you suspect to be a NE.</p>

<p>2) In a game theory context Pareto optimality means that there is no other outcome of the game where someone is better off and no one is worse off. Usually highly asymmetric payoffs are Pareto-optimal because someone is very happy, so you should check some asymmetric strategies.</p>

<p>3) The NE actually does not need enforcement. It is what players will play anyway. My guess is that your question was: How much do you need to tax cow placement to enforce the Pareto-optimal outcome as a NE of the taxed game?</p>

<p>Hope these will help YOU finish the problem. </p>

CC BY-SA 3.0
2015-01-26T19:05:12.183Z
Homework questions are still not really clear for us, but I think the last minimal consensus was shown effort.
FooBar
CC BY-SA 3.0
2015-01-26T18:48:59.753Z
2015-02-16T07:33:49.860Z
2015-02-16T07:33:49.860Z
Giskard
<p>Without any loss of generality, we will only consider profiles <span class="math-container">$(q_1,q_2,\ldots, q_5)$</span> that satisfy <span class="math-container">$q_1\geq q_2\geq q_3\geq q_4 \geq q_5$</span>. This is because, by symmetry, a profile <span class="math-container">$(q_1,q_2,\ldots, q_5)$</span> is Pareto optimal if and only if any permutation of it is Pareto optimal.</p>
<p>Check that any profile <span class="math-container">$(q_1,q_2,\ldots, q_5)$</span> that satisfy <span class="math-container">$2=q_1\geq q_2\geq q_3\geq q_4 \geq q_5=0$</span> is Pareto optimal. This is because changing the profile by increasing <span class="math-container">$\sum_{i=1}^{5} q_i$</span> will necessarily make individual 1 worse off. On the other hand, reducing or leaving <span class="math-container">$\sum_{i=1}^{5} q_i$</span> unchanged, will make either someone worse off or leave everyone as exactly as well off as they were before.</p>
<p>Likewise, any profile <span class="math-container">$(q_1,q_2,\ldots, q_5)$</span> that satisfy <span class="math-container">$2=q_1\geq q_2\geq q_3\geq q_4 = q_5=1$</span> is also Pareto optimal.</p>
<p>Finally, any profile <span class="math-container">$(q_1,q_2,\ldots, q_5)$</span> that satisfy <span class="math-container">$1=q_1= q_2= q_3= q_4 \geq q_5$</span> is also Pareto optimal. Check these yourself.</p>

CC BY-SA 4.0
2024-10-12T23:33:59.607Z
2024-10-12T23:42:37.423Z
2024-10-12T23:42:37.423Z
Amit
<p>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?) </p>

<p>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. </p>

<p>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. </p>

<game-theory><nash-equilibrium><pareto-efficiency>
Pareto optimality and Externalities
CrishaD
# Pareto optimality and Externalities

(Asked by: username_0 on 2015-01-26)

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.

**Tags:** game-theory, nash-equilibrium, pareto-efficiency

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## 3 answers

### Answer by username_1 on 2015-01-26. Score: 2

The idea behind Pareto optimality is *efficiency*. On the discrete set, there is no allocation where, compared to $(1,1,1,1,1)$, we could make someone better off and keep the others at least indifferent. This is why this allocation is Pareto optimal. However, any other *efficient* allocation (in the sense that we are not throwing away resources) will be Pareto optimal here as well.

For the set in $\mathcal R$, we assume a symmetric solution and solve

$$\max\_{q\in[0,2]} 5\cdot q(12-(5q))$$
$$\Rightarrow 5(12-5q) -25q$$

The first term is the direct benefit of increasing $q$, the second one is the externality (that we now internalize). The interior solution is given by $q = 1.2$ where everyone gets the payoff $7.2$. We know that the interior solution solves the problem given that the boundaries $0, 2$ both delivered lesser value than $q=1$.

Finally, for the tax, derive the FOC of the agents given a general tax $c$. Then, look for the optimal quantity *you want them to have*, that is, replace $q\_i$ with $q\_i^\*$, the welfare-maximizing solution, and then look for the value of $c$ that would solve the FOC (and hence make the farmers optimally chose what you want them to).


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### Answer by username_2 on 2015-01-26. Score: 0

My understanding is that we do not answer homework problems here. (I might be wrong.) However you did lay out some ideas and I will give you feedback on those:

1) There is no guarantuee that the NE will remain the same if you extend the strategy space. You have to check again by looking for the arg max of the payoff function given the strategy profile you suspect to be a NE.

2) In a game theory context Pareto optimality means that there is no other outcome of the game where someone is better off and no one is worse off. Usually highly asymmetric payoffs are Pareto-optimal because someone is very happy, so you should check some asymmetric strategies.

3) The NE actually does not need enforcement. It is what players will play anyway. My guess is that your question was: How much do you need to tax cow placement to enforce the Pareto-optimal outcome as a NE of the taxed game?

Hope these will help YOU finish the problem.

> **Comments:**
> **username_1**: Homework questions are still not really clear for us, but I think the last minimal consensus was shown effort.  

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### Answer by username_3 on 2024-10-12. Score: 0

Without any loss of generality, we will only consider profiles $(q\_1,q\_2,\ldots, q\_5)$ that satisfy $q\_1\geq q\_2\geq q\_3\geq q\_4 \geq q\_5$. This is because, by symmetry, a profile $(q\_1,q\_2,\ldots, q\_5)$ is Pareto optimal if and only if any permutation of it is Pareto optimal.

Check that any profile $(q\_1,q\_2,\ldots, q\_5)$ that satisfy $2=q\_1\geq q\_2\geq q\_3\geq q\_4 \geq q\_5=0$ is Pareto optimal. This is because changing the profile by increasing $\sum\_{i=1}^{5} q\_i$ will necessarily make individual 1 worse off. On the other hand, reducing or leaving $\sum\_{i=1}^{5} q\_i$ unchanged, will make either someone worse off or leave everyone as exactly as well off as they were before.

Likewise, any profile $(q\_1,q\_2,\ldots, q\_5)$ that satisfy $2=q\_1\geq q\_2\geq q\_3\geq q\_4 = q\_5=1$ is also Pareto optimal.

Finally, any profile $(q\_1,q\_2,\ldots, q\_5)$ that satisfy $1=q\_1= q\_2= q\_3= q\_4 \geq q\_5$ is also Pareto optimal. Check these yourself.


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