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Find the non-randomized Nash Equilibria of this two-player strategic game, in which each player's set of actions is the non-negative real numbers, and the players' payoff functions are:

$$u_{1} (c_{1},c_{2}) = c_{1} (c_{2}-c_{1})$$

$$u_{2} (c_{1},c_{2}) = c_{2} (1 - c_{1}-c_{2})$$

How does one approach this problem without a payoff matrix?

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  • $\begingroup$ Here is a hint: Every player needs to maximize their utility. So, try to determine the maximum $u_i$ with respect to $c_i$. $\endgroup$
    – Max
    Mar 18 at 4:14

1 Answer 1

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Solution:

  • Set of players: $\{1,2\}$
  • Action sets: $A_1 = \mathbb{R^+}$, $A_2 = \mathbb{R^+}$
  • The payoffs are,

$u_1(c_1,c_2)= c_{1} (c_{2}-c_{1})$

and, $u_2(c_1,c_2)= c_{2} (1 - c_{1}-c_{2})$

We can figure out the Best Response Set of player 1 and 2 and then look the intersection of the Graphs of Best responses to figure out the Nash Equilibria,

$BR_1(c_2)= \{c_1\in A_1 | u_1(c_1,c_2) \ge u_1(c_1^1,c_2) \hspace{2mm}, \forall c_1^1 \in A_1\}$ and similarly $BR_2(c_1)$

Figuring out the Best Response of Player 1:

$\max_{c_1} \hspace{2 mm} u_1(c_1,c_2)= \max_{c_1} \hspace{2mm} (c_1c_2-c_1^2)$ (Notice that function $u_1$ is concave in $c_1$, therefore the First order condition is both necessary and sufficient)

The First order condition is,

$$c_2-2c_1=0 \\ \Rightarrow c_2=2c_1$$ The Best Response of player 1 as a function of $c_2$ is $BR_1(c_2)=\{c_1 \in A_1 | c_2=2c_1\}$

Figuring out the Best Response of Player 2:

$\max_{c_2} \hspace{2 mm} u_2(c_1,c_2)= \max_{c_2} \hspace{2mm} (c_2-c_1c_2-c_2^2)$ (Notice that function $u_2$ is concave in $c_2$, therefore the First order condition is both necessary and sufficient)

The First order condition is,

$$1-c_1-2c_2=0 \\ \Rightarrow c_1=-2c_2+1$$ The Best Response of player 2 as a function of $c_1$ is $BR_2(c_1)=\{c_2\in A_2 | c_1=-2c_2+1\}$

Now we can plot the $GR(BR_1(c_2))=\{(c_1,c_2)\in A_1 \times A_2 |c_1 \in BR_1(c_2)\}$ and $GR(BR_2(c_1))=\{(c_1,c_2)\in A_1 \times A_2 |c_2 \in BR_2(c_1)\}$ and look at their intersection to figure out the equilibrium, which will be equivalent to solving, $$ c_2=2c_1 \\ 1-c_1-2c_2=0$$ which will give us the Nash-Equilibrium as, $(c_1^\ast,c_2^\ast)=(\frac{1}{5},\frac{2}{5})$

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