This document discusses two-person, zero-sum games and equilibrium points. It provides examples of games with equilibrium strategies and equilibrium points, where neither player has incentive to deviate from their equilibrium strategy. When equilibrium points exist, they represent stable solutions. The document also discusses dominated strategies that can be eliminated, and cases without equilibrium points where game theory is needed to play rationally.

1. SET 2: TWO-PERSON,
ZERO-SUM GAME WITH
EQUILIBRIUM POINTS
Erwin Widodo

2. INTRODUCTORY PROBLEMS
Game 1 Your opponent
I II III
You
A 5 -2 1
B 6 4 2
C 0 7 -1

3. INTRODUCTORY PROBLEMS
Game 2 Your opponent
I II III
You
A -2 1 1
B -3 0 2
C -4 -6 4

4. INTRODUCTORY PROBLEMS
Game 3 Your opponent
I II III IV
You
A -3 17 -5 21
B 7 9 5 7
C 3 -7 1 13
D 1 -19 3 11

5. INTRODUCTORY PROBLEMS
Game 4 Your opponent
I II III
You
A 2 -5 -2
B 3 -1 -1
C -3 4 -4

6. RULES
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
Your opponent
I II III
Yo
u
A 5 -2 1
B 6 4 2
C 0 7 -1

7. 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?

8. WHAT ABOUT THIS PAYOFF?
Game 5: missing payoff Your opponent
I II III
You
A ? ? 3
B ? ? 4
C 7 6 5

9. 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

10. 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

11. ITS PAYOFF MATRIX
Japanese choice
North South
Allies
choice
North 2 days 2 days
South 1 day 3 days

12. 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…

13. 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!

14. 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.

15. 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.

16. 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.

17. 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.

18. ITS PAYOFF MATRIX
B’s platform
Favor X Favor Y Dodge
issue
A’s
platfor
m
Favor
X 45% 50% 40%
Favor
Y 60% 55% 50%
Dodge
issue 45% 55% 40%

19. 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.

20. 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.

21. ITS MODIFIED MATRIX
B’s platform
Favor X Favor Y Dodge
issue
A’s
platfor
m
Favor
X 45% 10% 40%
Favor
Y 60% 55% 50%
Dodge
issue 45% 10% 40%

22. 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!

23. ITS RE-MODIFIED MATRIX
B’s platform
Favor X Favor Y Dodge
issue
A’s
platfor
m
Favor
X 35% 10% 60%
Favor
Y 45% 55% 50%
Dodge
issue 40% 10% 65%

24. 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.

25. 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.

26. 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)

27. 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.

28. 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.

29. ITS GT GRAPHICAL ANALYSIS
If the minimax equals the maximin, the payoff is an
equilibrium point  the corresponding strategies are
an equilibrium strategy pair.
B’s platform
Favor X Favor Y Dodge
issue
A’s
platfor
m
Favor
X 35% 10% 60%
Favor
Y 45% 55% 50%
Dodge
issue 40% 10% 65%
45 is the largest value in the
45 is the
smallest value
in the row
Equilibrium
point: the
match of the
smallest in
row and the
largest in
column

30. SOLUTION
When an equilibrium point exists in a two-person, 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

31. 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

32. 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
Favor X Favor Y Dodge
issue
A’s
platform
Favor X 35% 10% 60%
Favor Y 45% 55% 50%
Dodge
issue 40% 10% 65%

33. 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

34. ILLUSTRATING PAYOFF MATRIX
Your opponent
I II III
You
A 7 9 8
B 9 10 12
C 8 8 8

35. 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.

36. 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

37. CONSIDER THE FOLLOWING
MATRIX
Your opponent
I II III
You
A 19 0 1
B 11 9 3
C 23 7 -3

38. 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.

39. LET’S GO BACK TO FEB 1943
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
Japanese choice
North South
Allies
choice
North 2 days 2 days
South 1 day 3 days

40. LET’S GO BACK TO POLITICAL
EXAMPLE
“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
B’s platform
Favor X Favor Y Dodge
isssue
A’s
platfor
m
Favor
X 35% 10% 60%
Favor
Y 45% 55% 50%
Dodge
issue 40% 10% 65%

41. NO EQUILIBRIUM POINT
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)
Opponent
Heads Tails
Yo
u
Head
s -1 +1
Tails +1 -1

42. 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.