Set 2: Two-person, Zero-sum
game with equilibrium points
Erwin Widodo
Introductory problems
• Game 1
Your opponent
You
I
II
III
A
5
-2
1
B
6
4
2
C
0
7
-1
Introductory problems
• Game 2
Your opponent
You
I
II
III
A
-2
1
1
B
-3
0
2
C
-4
-6
4
Introductory problems
• Game 3
Your opponent
I
II
III
IV
A
-3
17
-5
21
B
7
9
5
7
C
3
-7
1
13
D
1
-19
3
11
You
Introductory problems
• Game 4
Your opponent
You
I
II
III
A
2
-5
-2
B
3
-1
-1
C
-3
4
-4
Your opponent
Rules
You
I
II
III
A
5
-2
1
B
6
4
2
C
0
7
-1
• Pick a row (A, B, or C) and your opponent picks a
column (I, II, or III) at the same time  neither
knows when choosing what the other has picked
• The number where the column and row intersect
is the amount your opponent pays you.
• In Ex 1: If you pick A and your opponent picks III,
you will get 1
• Assume that your opponent knows the rules and
as intelligent as you
• You must consider what your opponent is
thinking
Questions
•
•
•
•
What would you do?
Why?
What should the outcome of this game be?
If your choice depends on your opponent’s
choice, how do you play when you don’t know
what he/she will do?
What about this payoff?
• Game 5: missing payoff
Your opponent
You
I
II
III
A
?
?
3
B
?
?
4
C
7
6
5
Problem in Feb 1943
• Gen. George Churchill Kenney, Air Force
Commander in Southwest Pacific had a
problem.
• Japanese were about to reinforce their army
in New Guinea and had two alternative
routes:
– Sail north of New Britain with rainy weather
– Sail south of New Britain with fair weather
• In any case the journey takes 3 days
Problem in Feb 1943
• Gen Kenney had do decide where to concentrate
his spy aircraft.
• The Japanese wanted their ships to have the least
exposure to enemy.
• Of course Gen. Kenney wanted the reverse.
• The following matrix represents the expected
number of days of bombing exposure
Its payoff matrix
Japanese choice
Allies
choice
North
South
North
2 days
2 days
South
1 day
3 days
The difference
• A more difficult game than previous one.
• The critical different is here the players lack
information.
• Both players must decide simultaneously, so
neither knows the other strategy when
choosing his own.
• However, the analysis is simple…
The analysis
• The Allies thought it would be best for them to
take the same route as the Japanese.
• But when they made decision, they did not know
what the route would be.
• Nonetheless the problem would be solved when
they took the Japanese standpoint.
• For the Japanese, the northern route minimized
their exposure whatever the Allies did.
• So after working this out, its is clear, the Allies
decision was: Go North!
GT analysis
• The last example is an two-person, zero-sum game with
equilibrium points
• The term zero-sum (equivalently, constant sum) means the players
have diametrically opposed (=sangat bertentangan) interests.
• The term comes from parlor games like poker where there is fixed
money around the table.
• If you want to win some money  others have to lose an
equivalent amount.
• Please contrast with a trading between two nations (both may
simultaneously gain)
• An equilibrium point is a stable outcome of a game associated with
a pair of strategy.
• It is considered stable because a player unilaterally (affecting only 1
side) picking a new strategy is hurt by the change.
A political example
• This year is an election year and 2 major
political parties are busy in writing their
platforms (=janji2 politik)
• There is a dispute in district X and Y
concerning certain water rights.
• Each party must decide whether it will favor X
or favor Y or evade the issue.
• The parties will announce their decisions
simultaneously.
A political example
• Citizens outside the two states are indifferent to
the issue.
• In X and Y, the voting behavior of the electorate
(=pemilih) can be predicted from the past
experience.
• The regulars will support their party in any case.
• Others will vote for the party supporting their
state, or, if both parties take the same position on
the issue, will simply abstain.
A political example
• The leaders of both parties calculate what will
happen in each circumstance and come up
with the following payoff matrix.
• The entries are percentage of votes party A
will get if each party follows the indicated
strategy.
• Ex: if A favors X and B dodges (=mengelak) the
issue, A will get 40% of the vote.
Its payoff matrix
B’s platform
A’s
platform
Favor X
Favor Y
Dodge
issue
Favor X
45%
50%
40%
Favor Y
60%
55%
50%
Dodge
issue
45%
55%
40%
Its analysis
• This is the simplest example of this type of
game.
• Though both parties “have a hand” in
determining how the electorate will vote,
there is no point in one party trying to
anticipate what the other will do.
• Whatever A does, B does the best to dodge
the issue; Whatever B does, A does the best to
support Y.
Its analysis
• The predictable outcome is an even split.
• If, for some reason, one of the parties
deviated from the indicated strategy, this
should have no effect on the other party’s
actions.
• A slightly more complicated situation arises if
the percentages are changed a little as
follows.
Its modified matrix
B’s platform
A’s
platform
Favor X
Favor Y
Dodge
issue
Favor X
45%
10%
40%
Favor Y
60%
55%
50%
Dodge
issue
45%
10%
40%
Its following analysis
• B’s decision is now a bit harder.
• If B thinks A will favor Y, B should dodge the issue;
otherwise, B should favor Y.
• But the answer to the problem is in fact not far
off.
• A’s decision is clear-cut and easy for B to read:
favor Y.
• Unless A is foolish, B should realize that the
chance of getting 90% of the vote is very slim
indeed, and that it would do best: to dodge the
issue!
Its re-modified matrix
B’s platform
A’s
platform
Favor X
Favor Y
Dodge
issue
Favor X
35%
10%
60%
Favor Y
45%
55%
50%
Dodge
issue
40%
10%
65%
Its re-modified analysis
• Neither player has an obviously superior
strategy  both players must think a little.
• Each player’s decision hangs on what he
expects the other will do.
– If B dodges the issue, A should too.
– If not, A should favor Y.
– On the other hand, if A favor Y, B should favor X.
– Otherwise, B should favor Y.
Its GT analysis
• A is favoring Y and B is favoring X are important
enough to be given a name: Equilibrium Strategies.
• The outcome resulting from the use of these strategies
– the 45% vote for A – is called an Equilibrium point.
• Two strategies are said to be equilibrium (they come in
pairs, one for each player) if neither player gains by
changing strategy unilaterally.
• The outcome (sometimes called payoff) corresponding
to this pair of strategies is defined as equilibrium point.
Its GT analysis
• As the name suggests, equilibrium points are very
stable (once a player settled, there is no reason to
leave it)
• If A knew in advance that B would favor X, A would still
favor Y
• Similarly, B would not change strategy if he knew A
would favor Y
• There may be more than 1 equilibrium point, but if
there is, they will all have the same payoff.
• In a two-person, non-zero-sum game, equilibrium
points need not have the same payoff (check later in
Prisoner’s dilemma)
Its GT analysis
• Assume that B knows A’s strategy in advance.
• Since B would choose the minimum payoff of any
row A choose, A should choose a strategy that
yield the maximum of these minima  this value
is called the maximin.
• It is the very least that A can be sure of getting.
• If A plays “favor X”, “favor Y”, and “dodge issue”,
these minima are 10, 45, and 10 respectively.
• Thus, the maximin is 45.
Its GT analysis
• Now imagine the rules are changed so that A
knows B’s strategy in advance.
• A would be expected to choose the maximum of
any column, so B should choose the column that
minimizes these maxima  this value is called as
minimax.
• This is the very optimistic that B can avoid.
• If B plays “favor X”, “favor Y”, and “dodge issue”,
these maxima are 45, 55, and 65 respectively.
• Thus, the minimax is 45.
Its GT graphical analysis
• If the minimax equals the maximin, the payoff is an
equilibrium point  the corresponding strategies
are an equilibrium strategy pair.
Equilibrium
point: the match
of the smallest in
row and theFavor X
largest in column
A’s
platform
B’s platform
Favor X
Favor Y
Dodge
issue
35%
10%
60%
Favor Y
45%
55%
50%
Dodge
issue
40%
10%
65%
45 is the largest value in the column
45 is the
smallest value
in the row
Solution
• When an equilibrium point exists in a twoperson, zero-sum game, it is called a solution.
• Rational players should adopt the equilibrium
strategies and the outcome should be the payoff
associated with the equilibrium point – the value
of the game.
• In previous game:
– The equilibrium strategies were “favor Y” for A and
“favor X” for B
– The value of the game was 45
The reason why
equilibrium points = solutions?
• By playing his equilibrium strategy, a player will get at
least the value of the game. In previous example, A
gets at least 45 whatever B does, if A plays “Favor Y”
• By playing his equilibrium strategy, an opponent can
stop a player from getting any more than the value of
the game. By playing “Favor X”, B can limit A’s payoff to
45 whatever A does
• Since the game is zero-sum, a player’s opponent is
motivated to minimize the player’s payoff. When A
gets 45, B gets 55; if A gets any more, it must be
because B obtained that much less
Note on equilibrium point
• In games with equilibrium points, payoffs that
are not associated with either equilibrium
strategy have no bearing on the outcome.
• If the 2 payoff of 10, payoff of 60 and 65 were
changed in any way, the player should pick the
old same equilibrium strategies and outcome.
B’s platform
A’s
platform
Favor X
Favor Y
Dodge
issue
Favor X
35%
10%
60%
Favor Y
45%
55%
50%
Dodge
issue
40%
10%
65%
Domination
• It is often possible to simplify a game by
eliminating dominated strategies.
• Strategy A dominates strategy B if a player’s
payoff with strategy A is:
– always at least as much as that of strategy B
(whatever other players do) and,
– at least some of the time actually better than
strategy B
Illustrating payoff matrix
Your opponent
You
I
II
III
A
7
9
8
B
9
10
12
C
8
8
8
Its analysis
• For you, strategy B dominates both strategies
A and C because it always yields a higher
payoff.
• Your opponent’s strategy I dominates
strategies II and III (recall your opponent
wants the payoff values small)
• Although your opponent doesn’t always do
better with I than with II and III, he always
does at least and sometimes does better.
Possible assumptions
in zero-sum game
• You will never pick a dominated strategy –
why pick a dominated strategy when you can
do at least as well using the strategy that
dominates it?
• Your opponent will never pick a dominated
strategy – and this for the same reason that
you won’t
Consider the following matrix
Your opponent
You
I
II
III
A
19
0
1
B
11
9
3
C
23
7
-3
Its analysis
• In this game none of your strategies are
dominated initially.
• However, since III dominates I for your opponent,
you can eliminate I from your consideration.
• With I eliminated, B dominates A and C.
• With A and C eliminated, III dominates II
• The only undominated strategies, B and III, make
up an equilibrium strategy pair and the value of
the game is 3.
Let’s go back to Feb 1943
Japanese choice
Allies
choice
North
South
North
2 days
2 days
South
1 day
3 days
• The northern route dominated the southern one for
Japanese.
• After eliminating the Japanese southern route, we
eliminated the Allies southern route for the same
reason.
• The equilibrium strategies were North for both
armies; the equilibrium point is 2 days
Let’s go back to political
example
B’s platform
A’s
platform
•
•
•
•
•
Favor X
Favor Y
Dodge isssue
Favor X
35%
10%
60%
Favor Y
45%
55%
50%
Dodge
issue
40%
10%
65%
“Favor X” dominated “Dodge issue” for B
“Favor Y” dominated everything for A
“Favor X” dominated “Favor Y” for B
The equilibrium strategies were “Favor Y” for A and “Favor X” for B
The equilibrium point was 45
No equilibrium point
Opponent
Heads
Tails
Heads
-1
+1
Tails
+1
-1
You
• Since no strategy is dominated and there is no
equilibrium
• It is hard to see how you can play such a game
rationally.
• However, there is a theory that enables you to
play such games intelligently. (next chapter)
Solutions to our previous games
Game 1: ES are B and III with EP is 2
Game 2: ES are A and I with EP is -2
Game 3: ES are B and III with EP is 5
Game 4: ES are B and III with EP is -1
In each of games 1 through 4, both players will not lose
even though they announce their strategies in advance
to the opponent
• Game 5: whatever the values of the missing payoffs,
you can be sure of getting 5 by playing C; your
opponent can be sure of losing no more than 5 by
playing III. Since neither one of you can enforce the
payoff of 5, this is a plausible outcome.
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