When players in a game find no advantage in choosing one pure strategy over another, they may choose to randomize their actions. This is the heart of a Mixed Strategy Nash Equilibrium (MSNE). The core idea is the indifference principle a player will only be willing to randomize between several pure strategies if the expected payoff from each of those strategies is exactly the same, given the opponent's strategy mix.

This principle turns the problem of finding an equilibrium into a solvable system of equations. Your goal isn't to maximize your own payoff directly. Instead, you choose your probabilities to make your opponent indifferent. If your opponent is indifferent, they have no incentive to deviate from their own mixed strategy. If both players do this simultaneously, the system is in equilibrium.

Let's use a simple 2x2 game to see this in action. Consider a game of "Matching Pennies." Player 1 wants to match Player 2, while Player 2 wants to mismatch. If both show Heads or both show Tails, Player 1 wins a point. If they show different faces, Player 2 wins a point.

Let's define the players' probabilistic strategy vectors. Player 1 chooses Heads with probability $p$ and Tails with probability $1-p$. Player 2 chooses Heads with probability $q$ and Tails with probability $1-q$.

To find the equilibrium, we first calculate Player 1's expected utility for each of their pure strategies, assuming Player 2 plays their mixed strategy. The expected utility is the sum of the payoffs for each outcome multiplied by the probability of that outcome occurring.

Now, we apply the indifference principle. Player 1 will only be willing to mix their strategies if their expected payoff is the same regardless of whether they choose Heads or Tails.

This result is key. To keep Player 1 guessing, Player 2 must play a perfectly random 50/50 strategy. Any other mix would allow Player 1 to gain an advantage by choosing one pure strategy over the other.

Now, we do the same for Player 2, based on Player 1's probability, $p$.

Setting them equal gives us Player 1's equilibrium mixing probability:

So, the unique Mixed Strategy Nash Equilibrium for Matching Pennies is for both players to play Heads or Tails with a 50% probability. The MSNE is expressed as the set of strategy profiles:

$( (1/2, 1/2), (1/2, 1/2) )$

We can visualize these strategic choices using a best-response correspondence. This is a graph that plots a player's optimal strategy (or strategies) for every possible strategy of their opponent.

For Player 1 in Matching Pennies:

• If Player 2 chooses Heads more than 50% of the time ($q > 1/2$), Player 1's best response is to always play Heads ($p=1$). Why? Because $E_1(\text{Heads}) = 2q - 1$ and $E_1(\text{Tails}) = 1 - 2q$. If $q > 1/2$, then $2q-1 > 1-2q$.
• If Player 2 chooses Heads less than 50% of the time ($q < 1/2$), Player 1's best response is to always play Tails ($p=0$).
• If Player 2 chooses Heads exactly 50% of the time ($q = 1/2$), Player 1 is indifferent. Any mix $p ∈ [0, 1]$ is a best response.

The same logic applies to Player 2's response to Player 1's choice of $p$. The Nash Equilibrium is the point where these best-response correspondences intersect. In this case, it's at $(p=1/2, q=1/2)$.

The same logic extends to larger N x N games, though the algebra becomes more complex. For a player to be willing to mix among a set of $k$ pure strategies, their expected utility must be equal across all $k$ of those strategies. This creates a system of $k-1$ independent equations.

For example, in a 3x3 game, if Player 1 is mixing three strategies (A, B, C) with probabilities $p_1, p_2, p_3$, Player 2 must choose their mixing probabilities ($q_1, q_2, q_3$) such that:

We would then do the same for Player 2 to find Player 1's mixing probabilities. Solving these systems of linear equations reveals the equilibrium probabilities for each player.

It is important to note that a strategy might be strictly dominated and will never be played in a mixed strategy. In such cases, its probability in the mix will be zero, effectively removing it from consideration and simplifying the problem.