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What does q=3/7q=3/7 mean in this game?

In this game, q=3/7q = 3/7 means that Player 2 chooses their pure strategy Right (R) with a probability of 3/73/7, and consequently chooses their pure strategy Left (L) with a probability of 4/74/7. At this specific mixture, Player 1 is completely indifferent between choosing R and L, receiving an expected payoff of 12/712/7 regardless of their choice. Together with Player 1's equilibrium mix, this completes the mixed-strategy Nash equilibrium for the displayed game.

Conditions

  • The game is the displayed 2x2 normal-form game.
  • Player 2 mixes between R and L with probabilities qq and 1−q1-q.
  • The equilibrium concept is mixed-strategy Nash equilibrium.

Reasoning, step by step

  1. Identify the value of qq solved from the indifference equation.
  2. Interpret qq as the probability of Player 2 choosing strategy R.
  3. Calculate 1−q1-q as the probability of Player 2 choosing strategy L.
  4. Verify that at this qq, Player 1's expected payoffs from R and L are equal.
  5. Conclude that this qq is Player 2's equilibrium mixing probability.

Example

The board shows q=3/7q = 3/7. This means Player 2 plays R with probability 3/73/7 and L with probability 4/74/7. Substituting back, Player 1's expected payoff from R is 3−3(3/7)=12/73 - 3(3/7) = 12/7, and from L is 4(3/7)=12/74(3/7) = 12/7, confirming Player 1 is indifferent.

Common misconceptions

  • One may mistakenly attach qq to Player 2's left action or to the wrong column payoff. In this worked example, qq is the probability of Player 2 choosing R.
  • Assuming that q=3/7q=3/7 is Player 1's probability of choosing R. It is specifically Player 2's mixing probability.

Watch the explanation

3:30 – 3:49Watch this moment ↗

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.