Guillermo Sabbioni
Game Theory (ECO4400)
Book: Games of Strategy, by Dixit & Skeath
Chapter 8: Simultaneous-move games with mixed strategies II: non-zero-sum games.
Mixing arises in non-zero sum game as well, even when there may not be conflicting
interests.
1. Mixing sustained by uncertain beliefs
Mixed strategies equilibria can arise in those games when players have subjectively
uncertain but correct beliefs.
a. Will Harry meet Sally?
See fig 8.1, the assurance game. If Sally is unsure about where Harry will go, then where
should she go? Subjective uncertainty: she thinks he is using a mixed strategy where p
is the prob that she thinks Harry will go to Starbucks. So how should she act? Compare
her payoffs from choosing her pure strategies against his p-mix. See fig 8.2. If she
chooses Starbucks her expected payoff is p and if she chooses LocalLatte it is 2(1-p).
Find her BR: see fig 8.3. If p=2/3 she gets the same expected payoff from her two pure
strategies. For p>2/3 she should go to Starbucks. See her BR in the right panel. Why not
½? Local Latte is more preferred by her! The subjective uncertainty in Sally’s mind
(about what Harry is doing) can lead to an objective uncertainty in her own action.
Calculate Harry’s BR in the same way. Symmetric payoffs, so identical BR with the axes
changed (see fig 8.4). Three NE. The mixed strategies NE is p=2/3 and q=2/3. Given
Sally’s belief about Harry’s p-mix, she is indiff between all of her strategies (pure and
mix). But only q=2/3 is sustainable as an equilibrium. If she does not choose q=2/3 then
Harry does not either (he could do better with a pure strategy). If they choose
independently they meet in Starbucks only (2/3)*(2/3)=4/9 of the time and 1/9 of the time
in Local Latte. Expected payoff: 2/3 (less than 1 that they would get if meeting at
Starbucks). Reason: they may fail to coordinate and not meet at all. If similar payoffs for
both from both cafes: p=1/2 and q=1/2 is the mixed strategies NE. In a mixed strategies
NE, each person’’s mixture keep the other player indiff between his pure strategies.
b. Diced chicken?
See fig 8.5. To find Dean’s BR consider that James chooses Swerve with prob p and
Straight with prob 1-p. See fig 8.6. Dean will be indiff between his pure strategies when
(p-1)=(3p-2), i.e. p=1/2. If p>1/2 (Swerve is likely from James) then Dean’s BR is
Straight. Graph of both BR in fig 8.7. Again, a lower payoff (-1/2) for both when they are
mixing independently (they can crash more often than coordinating).
2. Non-zero sum mixing with 3 strategies
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Guillermo Sabbioni
Expanded version of chicken: see fig 8.8. Four NE where one player goes Straight and
the other Swerves (R or L). Fully mixed equilibrium (using all strategies)? Set up p1 for
Left, p2 for Straight and 1-p1-p2 for Right. Now, Dean mixes when James p-mix keeps
him indiff among all his choices:
L=Straight: -p2-2(1-p1-p2) = p1-2p2+(1-p1-p2)
Straight=R: p1-2p2+(1-p1-p2) = -2p1-p2
(no need for L=R)
Solution of 2x2 system: p2=2/3, p1=1/6 and (1-p1-p2)=1/6 for Right.
Same for the q’s (symmetry again). Notice no negative probs so OK to use all pure
strategies. They use Straight (p2=2/3 and q2=2/3) more often! Swerving is not as safe
now (they could still crash if both swerve to the same side). But there are also “partially”
mixed NE (that is: not using all the pure strategies). If no R from Dean, for ex, then
James does not use R either. Also, it is possible that one chooses a pure strategy and the
other mixes… lots of cases.
3. General discussion
We introduce general properties of mixed strategies equilibria.
a. Weak sense of equilibrium
Critic: if a player is indiff between his pure strategies (when the other plays the right mix)
why would he choose his specific mix as a BR? Because if not, the other would not
choose his mixture. It would not be a stable outcome.
b. Opponent’s indifference
In zero-sum games, choosing the right mix also prevents exploitation from the rival
(another reason to choose the right mixture).
c. Counterintuitive outcomes
Change payoffs in the tennis game: see fig 8.9 (Navratilova improved her DL defense
against DL by Evert: Evert only succeeds 30% of the time). Still no equilibrium in pure
strategies. Should Navratilova go DL more often since she is now better doing so? Her
equilibrium was q=0.6 before (60% of the time choose DL).Calculate Evert’s
indifference to find q=0.5 (lower than before!). Intuition: since Navratilova is better at
DL Evert will use less DL so actually Navratilova does not need to use more DL. Now
calculate Navratilova’s indifference: 30p+90(1-p)=80p+20(1-p) to find p=58.3% for
Evert (less DL as mentioned, because Navratilova is better covering it). So where do we
see Navratilova’s improvement? Her expected payoff is now 45 (before it was 38). Not
counterintuitive from this point of view. Lesson: improve DL so you have to use it less
often.
Read the football example on your own: figs 8.10 and 8.11 (starts on page 246).
d. Another counterintuitive outcome
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Guillermo Sabbioni
General 2x2 non zero sum game. See fig 8.12. Row plays Up with prob p. Make column
indiff: p*A+(1-p)*C = p*B+(1-p)*D. Solution: p=(D-C)/(A-B+D-C). It does not contain
Row’s payoffs (only those of the rival). The same for the right q-mix for Column: it only
contains Row’s payoffs. Counterintuitive? No: remember that your goal when choosing
your mix is to keep your opponent indifferent (then: consider only his payoffs).
4. Evidence of mixing in non-zero sum games.
Evidence is poor (read yourselves). Conclusion: mixed strategy equilibria in non-zero
sum games should be used with caution.
5. Mixing among 3 or more strategies (NO)
SUGGESTED EXERCISES: 1, 2, 3, 4, 6.
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