Find the Nash Equilibria in this Two-Player Strategic Game

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?

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

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, $$(c_1^\ast,c_2^\ast)=(\frac{1}{5},\frac{2}{5})$$