MG_L3

Question 1

What is a mixed strategy?

Answer 1

• A strategy where a player assigns probabilities to each pure action.
• For player i, let their pure strategies be S_i.
• A mixed strategy is any probability distribution over S_i.

Question 2

What is a Nash equilibrium?

Answer 2

• A profile of strategies (possibly mixed) where no player can unilaterally deviate and gain a higher payoff.
• Formally, each player’s chosen strategy is a best response to the others’ strategies.

Question 3

How do we find a mixed-strategy Nash equilibrium?

Answer 3

• Compute each player’s best response as a function of the others’ mixed strategies.
• Identify an intersection where all best responses coincide.
• Players are often indifferent among pure strategies used with positive probability.

Question 4

What is expected utility?

Answer 4

• The weighted average of utilities over all possible outcomes under a random strategy.
• For outcomes with utilities u₁, u₂, … having probabilities p₁, p₂, …
• Expected utility = Σ (p_k × u_k).

Question 5

What is a strictly dominated strategy?

Answer 5

• A strategy that yields a lower payoff than another strategy, for every possible action profile of the other players.
• Strictly dominated strategies are never part of a Nash equilibrium.

Question 6

What is a weakly dominated strategy?

Answer 6

• A strategy that never gives a higher payoff than another, and in at least one case gives a strictly lower payoff.
• A weakly dominated strategy can still appear in some Nash equilibria, unlike a strictly dominated one.

Question 7

What is iterated deletion of dominated strategies?

Answer 7

• Repeatedly remove strictly dominated strategies from the game.
• The remaining reduced game has the same set of Nash equilibria as the original.
• This process can simplify the search for equilibria.

Question 8

Explain the matching pennies equilibrium.

Answer 8

• Pure strategies (Heads or Tails) have no equilibrium.
• The mixed equilibrium is each player choosing Heads with probability 1/2 and Tails with probability 1/2.
• Each player’s expected payoff is 0 at equilibrium.

Question 9

What does Nash’s Theorem (1950) state?

Answer 9

• In a finite game with a finite number of players and strategies, there is at least one Nash equilibrium (possibly in mixed strategies).
• This was a foundational result in modern game theory.

Question 10

Why might professionals follow mixed-strategy predictions?

Answer 10

• Studies (e.g., penalty kicks in football) show experts use near-Nash mix proportions.
• They ensure opponents remain indifferent, preventing exploitation.
• Demonstrates that high stakes and experience can bring play closer to theoretical predictions.