Game theory problem, 3x3 matrix: pure and mixed strategies
From the following matrix: $$ \left( \begin{array}{c|ccc} & L & C & R\\ \hline T &3,0& 1,1 &4,2\\ M &3,4& 1,2 &2,3\\ B &1,3& 0,2 &3,0 \end{array} \right) $$
I have derived the following Nash Equilibrium: ($M$,$L$) and ($T$,$R$). But I cannot, it seems, solve it w.r.t. mixed strategies. I have used an embarrassing amount of time trying to do so. Could somebody please help me solve the matrix for mixed strategies?
Thanks in advance.
$\endgroup$ 11 Answer
$\begingroup$Here's one sensible sequence of steps:
Step 1: Notice that T strictly dominates $B$, since $(3,1,4)$ is componentwise strictly greater than $(1,0,3)$. Remove $B$ and we are left with a $2 \times 3$ game.
Step 2: In this new game, with $B$ removed, $R$ dominates $C$, since $(2,3)$ is componentwise strictly greater than $(1,2)$. After removing $C$ we are left with a $2 \times 2$ game:
$$ \left( \begin{array}{c|cc} & L & R\\ \hline T &3,0& 4,2\\ M &3,4& 2,3\\ \end{array} \right) $$
Step 3: Having found two pure equilibria already, look for non-pure equilibria. Player 2 can be made indifferent between $L$ and $R$ as we see below. But, player 1 cannot be made indifferent between $T$ and $M$ because $T$ weakly dominates $M$: as soon as there is any positive probability on $R$, player 1 strictly prefers $T$. Thus player 2 cannot mix in equilibrium, and actually the pure equilibrium $(M, L)$ is actually only the endpoint of a range of equilibria:
$$ ((1-p, p), L)\ \text{where } p \in [2/3, 1] $$
The threshold of $p=2/3$ is the point at which player II is indifferent between $L$ and $R$ against $(1-p,p)$. When $p=2/3$ both L and R give expected payoff $1/3 \cdot 0 + 2/3 \cdot 4 = 1/3 \cdot 2 + 2/3 \cdot 3 = 8/3$.
A range of equilibria like this is only possible is a degenerate game. A $2 \times 2$ game is necessarily degenerate. More generally, a game is degenerate if there exists a mixed strategy $x$ with support size $k$ (i.e. , $|\{x_i \ | \ x_i >0\}| = k$) and more than $k$ best responses against $x$. In this example, the strategy $x$ is the pure strategy $L$, which has support size 1 but two best responses against it, $TM$ and $M$.
You can check the equilibria at:
- ; or
The output from the former for this game is:
2 x 2 Payoff matrix A: 3 4 3 2
2 x 2 Payoff matrix B: 0 2 4 3
EE = Extreme Equilibrium, EP = Expected Payoff
Decimal Output EE 1 P1: (1) 0.333333333333 0.666666666667 EP= 3.0 P2: (1) 1.0 0.0 EP= 2.66666666667 EE 2 P1: (2) 1.0 0.0 EP= 4.0 P2: (2) 0.0 1.0 EP= 2.0 EE 3 P1: (3) 0.0 1.0 EP= 3.0 P2: (1) 1.0 0.0 EP= 4.0
Rational Output EE 1 P1: (1) 1/3 2/3 EP= 3 P2: (1) 1 0 EP= 8/3 EE 2 P1: (2) 1 0 EP= 4 P2: (2) 0 1 EP= 2 EE 3 P1: (3) 0 1 EP= 3 P2: (1) 1 0 EP= 4
Connected component 1:
{1, 3} x {1}
Connected component 2:
{2} x {2}Here one of the two equilibrium components (the range of equilibria described above in terms of $p$) is denoted by
Connected component 1:
{1, 3} x {1}The other equilibrium component is the pure strategy equilibrium $(T,R)$.
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