Mixed Strategy Nash Equilibria
David A. Hughes, Ph.D.
Auburn University at Montgomery [email protected]
Introduction
By the end of this unit, students should be able to:
• Define the concept of a mixed strategy Nash equilibrium, • Apply this concept to a variety of simultaneous and sequential
games, and
Mixed strategies
• Previously, we’ve considered games that have no Nash equilibrium in pure strategies.
• This scenario often emerges in games, say, that are zero sum. • Knowing what your opponent would do gives you a clear best
response, but that best response would result in your opponent changing strategies.
Matching pennies
• Consider this zero-sum game.
• Each player chooses a side of the coin to play.
• If the sides match, P1 wins, and so on.
Expected utility
• To solve games with mixed strategies, we’ll need to introduce the idea of “expected utility.”
• You can think of expected utility as the payoff one anticipates given some lottery (probability distribution) over discrete strategy profiles.
• For example, suppose a player has three potential payoffs, (10, 4, 0) and assigns a lottery, (12,14,14) to that payoff profile. • Then in expectation, her utility from the payoff function and
lottery would be EU = 10 · 12+ 4 · 14+ 0 ·14 = 6.
Mixed strategy Nash equilibria (MSNE)
• Let αi(ai) denote the probability a player assigns to a given
action such that every action is assigned some probability, and the sum of all probabilities is 1.
• And let Ui(α) denote a player’s expected utility for a given
mixed strategy profile, α.
• Then a MSNE is said to attain if:
Ui(α∗) ≥ Ui(α0i, α∗−i),
Matching pennies (cont’d.)
• Suppose players mixed such that P1 played H with
probability p and T with probability 1 − p, and P2
played H with probability q and T with probability 1 − q. • What is each player’s
expected utility from either of their available actions?
H(q) T (1 − q)
H(p) 1, −1 −1, 1
Matching pennies (cont’d.)
• From the previous slide, we can calculate P1’s expected utility from playing either of their actions:
EU1(H) = 1(q) − 1(1 − q)
= 2q − 1.
EU1(T ) = −1(q) + 1(1 − q)
= 1 − 2q. • And P2’s expected utility is:
EU2(H) = −1(p) + 1(1 − p)
= 1 − 2p.
EU2(T ) = 1(p) − 1(1 − p)
Best responses for the pennies game
• Suppose that P1 formed the belief that q < 12. Then her expected utility from T would always exceed that for H. • And the opposite would attain for any belief she formed such
that q > 12.
• We could form a similar analysis for P2.
• The only time either player’s best response is a mixed strategy is when αi = 12 where each player’s expected utility is 0, thus
Solving for MSNE
• Indeed, a MSNE is said to attain if every player can be made indifferent among their strategies.
• To find a point of indifference, set a player’s expected utility from their actions equal to one another.
• Let’s find the level of q that makes P1 indifferent between H
and T :
EU1(H) = EU1(T )
2q − 1 = 1 − 2q
q∗ = 1 2.
Solution to matching pennies
• The preceding slides show us a few things:
• There is no solution in pure strategies to games like matching pennies.
• The unique Nash equilibrium is a mixed strategy profile where each player randomizes between heads and tails with
probability 12.
The play-caller’s dilemma
• Recall this game of football where the offensive and defensive coordinator choose simultaneously.
• There’s no PSNE, so solve for the unique MSNE. • How can each player make
the other indifferent between their strategies?
Def.
B C
Off. R −1, 1 1, −1
Coordination games reconsidered
• We previously found 2 PSNE for the following type of coordination game. • What is the additional Nash
Mixing over three strategies
• Let α1 = (p, q, 1 − p − q). • Let α2 = (m, n, 1 − m − n). • Solve for the unique MSNE.
Solve using (mixed) iterative dominance
• No strategy here is dominated in pure strategies.
• Nevertheless, notice that for P1, the
mixture, α1 = (0, 1/4, 3/4) dominates
the pure strategy of U .
• With U eliminated, next note that the mixture, α2 = (0, 1/2, 1/2) dominates
the pure strategy of L.
Mixed strategies in the extensive form
• Find the set of PSNE, SPNE, and MSNE. • How do they differ, and
Conclusion
• In this section, we considered mixed strategy Nash equilibria. • We found that games that both have and don’t have PSNE
can also have MSNE.
