Mixed-Strategy Equilibria: Probabilistic Actions and Beliefs

Mixed Strategies as Probability Distributions

Let's start by formalizing what a mixed strategy is. As you've seen in the calculations, it's a set of probabilities assigned to a player's pure strategies.

As the text states, a mixed strategy for a player is simply a probability distribution over their set of actions. If Player 1 has actions {Up, Down}, a mixed strategy ${\sigma }_1$ is a function that assigns a probability to each, for instance, ${\sigma }_1(\text{Up})=p$ and ${\sigma }_1(\text{Down})=1-p$, where $0\le p\le 1$.

The payoffs we've been using are assumed to be Bernoulli payoffs, meaning a player's preference over lotteries (which is what a mixed strategy creates) can be represented by the expected value of these payoffs.

Given this formal definition, we can now explore our first and most direct interpretation.

Interpretation 1: Deliberate Randomization

The most straightforward way to understand a mixed strategy is that players are literally and intentionally randomizing their choices. A player with an equilibrium strategy ${\sigma }_1=(0.7,0.3)$ would, in effect, use a randomizing device—like a weighted coin or a computer program—that selects their first action 70% of the time and their second action 30% of the time.

This interpretation is most natural in purely confrontational, zero-sum games where being predictable is a significant disadvantage.

Mixed Strategies as Beliefs

Deliberate randomization makes sense for Matching Pennies, but what about coordination games like the Battle of the Sexes? In such games, the players want to coordinate, so actively trying to "confuse" each other seems counterproductive. This suggests we need a more nuanced interpretation.

Interpretation 2: Mutually Consistent Beliefs

In this view, a mixed strategy doesn't necessarily describe what a player does, but rather what their opponent believes they will do.

• Player 1's mixed strategy, ${\sigma }_1$ with probability $p$ for 'Up', represents Player 2's belief that Player 1 will choose 'Up' with probability $p$.
• Player 2's mixed strategy, ${\sigma }_2$ with probability $q$ for 'Left', represents Player 1's belief that Player 2 will choose 'Left' with probability $q$.

A mixed-strategy Nash equilibrium is then a state of mutually consistent beliefs. My belief about you justifies my actions, and your belief about me justifies your actions.

To formalize this, we can define a belief as a probability measure over the other players' actions.

Now, let's see how this "belief" interpretation works in practice.

The Unifying Role of the Indifference Principle

How can a mixed strategy be both a distribution of actions and a system of beliefs? The two interpretations are linked by the very indifference principle you used for your calculations.

Let's reason from Player 1's perspective, holding a belief ${\sigma }_2$ about Player 2's actions.

• If Player 1's belief ${\sigma }_2$ leads to a higher expected payoff for their 'Up' action (i.e., ${\mathbb{E}}_1[\text{Up}|{\sigma }_2]>{\mathbb{E}}_1[\text{Down}|{\sigma }_2]$), they will play 'Up' with certainty. Their best response is a pure strategy.
• It's only when their belief ${\sigma }_2$ is precisely the one that makes them indifferent (${\mathbb{E}}_1[\text{Up}|{\sigma }_2]={\mathbb{E}}_1[\text{Down}|{\sigma }_2]$) that they are willing to play a mixed strategy themselves. Any mix becomes a best response.

Therefore, for Player 1 to play a mixed strategy, their belief about Player 2 must be the exact equilibrium strategy ${\sigma }_2^{\ast }$. Symmetrically, for Player 2 to play a mixed strategy, their belief about Player 1 must be ${\sigma }_1^{\ast }$.

An MSNE is the unique point where Player 1's chosen probability distribution ${\sigma }_1^{\ast }$ is precisely the belief Player 2 needs to hold to be willing to play ${\sigma }_2^{\ast }$, and vice-versa. The beliefs and actions must be consistent.

The following proposition from Osborne's text formalizes this logic. It states that in an MSNE, a player must be indifferent between all pure strategies they play with positive probability. This indifference is induced by their beliefs about the other players' strategies.

Interpretation 3: Population Frequencies

A third, powerful interpretation emerges when we think about games played not by two individuals, but by large populations. This viewpoint is central to evolutionary game theory and has strong parallels with concepts like steady states in statistical mechanics.

In this interpretation:

• An equilibrium probability like $p^{\ast }=0.7$ for 'Up' doesn't mean every individual is randomizing.
• Instead, it could mean that in a large population of 'Player 1s', 70% are of a "type" that always plays 'Up', and 30% are of a "type" that always plays 'Down'.

When a player from the 'Player 2' population is randomly matched with a player from the 'Player 1' population, they face an opponent who will play 'Up' with a 70% probability. The MSNE describes a stable proportion of these deterministic strategies in the population.

Conclusion

You have now moved beyond the mechanics of mixed strategies to understand their deeper meaning. The probabilities in a mixed-strategy Nash equilibrium are not just numbers; they are a rich concept that can be interpreted in several ways, depending on the context of the game.

Key Takeaways:

• A mixed strategy is formally a probability distribution over a player's pure actions.
• Interpretation 1: Deliberate Randomization. Players literally use randomizing devices to be unpredictable, which is crucial in zero-sum or confrontational games.
• Interpretation 2: Mutually Consistent Beliefs. A player's mixed strategy represents their opponents' beliefs about their actions. An MSNE is a stable state where every player's belief about others makes them willing to behave in a way that confirms those beliefs.
• Interpretation 3: Population Frequencies. In large populations, a mixed strategy can represent the proportion of players who are 'hard-wired' to play each pure strategy. The MSNE is a steady-state distribution of types.

In our next lesson, we will put these concepts into practice by applying the calculation and interpretation of mixed strategies to two classic games: Matching Pennies and Battle of the Sexes. This will solidify your understanding of how the context of a game guides our interpretation of its equilibrium.