Introduction
Hello! In our last lesson, we explored the Stag Hunt game and the strategic challenges that arise when a game has multiple pure-strategy Nash equilibria. We saw that concepts like payoff dominance and risk dominance help us analyze which outcome might be more likely, but the choice ultimately hinges on trust and coordination.
In every game we've examined so far, from the Prisoner's Dilemma to the Stag Hunt, we've always found at least one pure-strategy Nash equilibrium. This leads to a natural and critical question: what if a game has no pure-strategy Nash equilibrium at all? What does "rational" play look like when every outcome is unstable, and someone always wants to switch their strategy?
This lesson addresses that exact problem. We will uncover the rationale for a new, expanded concept of strategy: the mixed strategy. You will learn why randomizing your actions is not just a sign of indecision, but can be the only truly strategic way to play in certain kinds of games.
The Problem: When Best Responses Go in Circles
To understand the need for a new solution concept, let's analyze a game where our current tools fall short. The classic example is a simple game called Matching Pennies.
In this game, two players each place a penny on a table, either Heads (H) or Tails (T).
- If the pennies match (H-H or T-T), Player 1 wins and gets a payoff of 1, while Player 2 loses and gets -1.
- If the pennies don't match (H-T or T-H), Player 2 wins and gets 1, while Player 1 loses and gets -1.
This is a pure conflict, or zero-sum, game. Let's try to find a pure-strategy Nash Equilibrium using best response analysis on the payoff matrix:
| P2: Heads | P2: Tails | |
|---|---|---|
| P1: Heads | (1, -1) | (-1, 1) |
| P1: Tails | (-1, 1) | (1, -1) |
- If P2 plays Heads, P1's best response is Heads (1 > -1).
- If P2 plays Tails, P1's best response is Tails (1 > -1).
- If P1 plays Heads, P2's best response is Tails (1 > -1).
- If P1 plays Tails, P2's best response is Heads (1 > -1).
As you can see, the best responses never align in a single cell. No matter which outcome we land on, one player will always regret their choice and wish they had played differently. This game has no pure-strategy Nash Equilibrium.
1. Mixed Strategy Nash Equilibrium (Game Theory Playlist 4)
The following video from a game theory course by Selcuk Ozyurt provides a concise walkthrough of this exact problem, showing why games like Matching Pennies demand a new kind of solution.
Starting from the beginning for about three minutes, watch the introductory analysis. The video will reinforce the best response analysis we just did and highlight why the absence of a pure-strategy equilibrium is a problem for our theory.
This presents a puzzle. If we assume players are rational, what should they do? If Player 1 predictably plays Heads, Player 2 will exploit this by playing Tails. But if Player 2 starts playing Tails, Player 1 should switch to Tails. This cycle of responses means there is no stable, predictable outcome.
The Solution: Strategic Unpredictability
The key insight is that in a game like Matching Pennies, predictability is a liability. The only way to avoid being exploited is to be unpredictable. This is achieved through randomization.
[PDF] 1 Describing Mixed Strategy Nash Equilibria
This document from a course at UNC Charlotte gives a clear, intuitive explanation for why players must randomize in games without a pure-strategy equilibrium.
Please read Section 1.1, 'Playing strategies with probabilities'. Starting just below the heading, read about mixed strategies. Focus on the logic of why always choosing one action (a pure strategy) cannot be part of an equilibrium in this type of game.
This leads us to the formal concept of a mixed strategy.
A mixed strategy for a player is a probability distribution over their set of pure strategies. Instead of choosing a single action, the player chooses which probabilities to assign to each action, and then lets a randomizing device (like flipping a coin or rolling a die) pick the action for them.
For example, in Matching Pennies, a mixed strategy for Player 1 could be:
- Play Heads with probability
- Play Tails with probability
Note that a pure strategy is just a special case of a mixed strategy where or . By allowing for mixed strategies, we are expanding the set of possible actions for players, which, as John Nash proved, guarantees that an equilibrium will always exist.
The Indifference Principle: Why a Specific Mix?
So, players should randomize. But with what probabilities? Should Player 1 play Heads 70% of the time? 20%? 50%? What is the strategic rationale for choosing a particular probability mix?
The answer lies in a beautiful and counter-intuitive piece of logic called the indifference principle.
The goal of your mixed strategy is to choose probabilities that make your opponent indifferent between their pure strategies.
Let's unpack this. In Matching Pennies, if Player 1 plays H with probability and T with probability , what is Player 2's best response? Player 2 will calculate their expected payoff from playing H versus playing T.
- P2's expected payoff from playing H:
- P2's expected payoff from playing T:
If , Player 2 will play H. If , Player 2 will play T. In either case, Player 2 has a single best response that Player 1 can then be exploited by.
But what if Player 2 is indifferent? That is, what if their expected payoffs are equal?
When Player 1 plays Heads and Tails with a 50/50 split, Player 2's expected payoff from playing Heads is 0, and their expected payoff from playing Tails is also 0. Since Player 2 is indifferent, they have no strict preference for one action over the other. Any mix of H and T is a best response for Player 2. By making Player 2 indifferent, Player 1 prevents Player 2 from finding a single, optimal counter-strategy to exploit.
This is the core rationale for mixed strategies.
9. Mixed strategies in theory and tennis
Professor Ben Polak from Yale University explains this 'indifference principle' masterfully. He calls it the 'enormous consequence' and 'the trick' to finding mixed strategy equilibria.
Watch the section on the indifference principle. Focus on the core argument: if a player is mixing strategies in a best response, they must be indifferent between the pure strategies they are mixing over. The analogies about picking the tallest TA or the student with the highest GPA are particularly insightful.
A Mixed Strategy Nash Equilibrium (MSNE) is a profile of mixed strategies, one for each player, such that no player can get a higher expected payoff by unilaterally changing their own mixed strategy. This stability is achieved precisely because each player's mix makes the other players indifferent among the actions they are using in their own mix.
Conclusion
Today, we confronted a fundamental limitation of pure strategies and discovered the elegant solution that game theory provides.
Key Takeaways:
- Some games, like Matching Pennies, have no pure-strategy Nash Equilibrium. In these games, players always have an incentive to deviate from any predictable outcome.
- The solution to this instability is for players to be unpredictable by randomizing their actions using mixed strategies.
- A mixed strategy is a probability distribution over a player's set of pure strategies.
- The core rationale for choosing specific probabilities in a mixed strategy is the indifference principle: you randomize in such a way as to make your opponent indifferent between the actions they are choosing from. This removes their ability to find a single, exploitative counter-move.
We have now established the why behind mixed strategies. In our next lesson, we will build directly on the indifference principle to master the how: you will learn the step-by-step mathematical method for calculating the probabilities that form a mixed strategy Nash equilibrium in 2x2 games.