Two Player Zero Sum Games

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Two Player Zero Sum Games

• Virtual Material for Statistics 802

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The General (m by n) Two Player, Zero Sum Game

• 2 players
• opposite interests (zero sum)
• communication does not matter
• binding agreements do not make sense

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The General Two Player Zero Sum Game

• Row has m strategies
• Column has n strategies
• Row and column select a strategy simultaneously
• The outcome (payoff to each player) is a function
  of the strategy selected by row and the strategy
  by column
• The sum of the payoffs is zero (zero-sum,
  remember)

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Sample Game Matrix

• Column pays row the amount in the cell
• Negative numbers mean row pays column

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2 by 2 Example

• Row collects some amount between 14 and 67 from
  column in this unfair game (The game is unfair
  because column can not win.)

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2 by 2 Example Row Interchange

• Rows, columns or both can be interchanged without
  changing the structure of the game. In the two
  games below Rows 1 and 2 have been interchanged
  but the games are identical!!

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2 by 2 representations (Each player has 2
strategies)

• Each 2 by 2 game has 4 representations
• original
• interchange rows
• interchange columns
• interchange rows and columns

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Simple Games - 1Rows choice
Reminder Column pays row the amount in the
chosen cell.
You are row. Should you select row 1 or row 2 and
why? Remember, row and column select
simultaneously.
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Simple Games - 1Rows Answer
Reminder Column pays row the amount in the
chosen cell.
You are row. Should you select row 1 or row 2 and
why? Remember, row and column select
simultaneously. You should select row 2 because
regardless of which column is chosen row 2 is
better. If column selects col 1 then row 2 yields
34 instead of only 11 while if column selects
col 2 row 2 yields 42 instead of only 27. Row
wants to collect as much as possible.
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Simple Games - 1Columns choice
Reminder Column pays row the amount in the
chosen cell.
You are column. Should you select col 1 or col 2
and why? Remember, row and column select
simultaneously.
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Simple Games - 1Columns Answer
Reminder Column pays row the amount in the
chosen cell.
You are column. Should you select col 1 or col 2
and why? Remember, row and column select
simultaneously. You should select col 1 because
regardless of which row is chosen col 1 is
better. If row selects row 1 then col 1 pays only
11 instead of 27 while if row selects row 2 col
2 pays only 34 instead of 42. Column wants to
pay as little as possible.
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Domination
Reminder Column pays row the amount in the
chosen cell.
We say that row 2 dominates row 1 since row 2 is
better regardless of which column is chosen.
Similarly, we say that column 1 dominates column
2 since each outcome in column 1 is better than
the corresponding outcome in column 2.
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Domination
Reminder Column pays row the amount in the
chosen cell.
We can always eliminate rows or columns which are
dominated in a zero sum game.
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Simple Games - 1Game Solution
Reminder Column pays row the amount in the
chosen cell.
Thus, we have solved our first game (and without
using DS for Windows.) Row will select row 2,
Column will select col 1 and column will pay row
34. We say the value of the game is 34. We
previously had said that this game is unfair
because row always wins. To make the game fair,
row should pay column 34 for the opportunity to
play this game.
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Simple games - 2
Answer the following 3 questions before going to
the following slides. What should row do? (easy
question) What should column do? (not quite as
easy) What is the value of the game (easy if you
got the other 2 questions)
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Simple games - 2Rows choice
As was the case before, row should select row 2
because it is better than row 1 regardless of
which column is chosen. That is, 55 is better
than 18 and 30 is better than 24.
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Simple games - 2Columns choice
Until now, we have found that one row or one
column dominates another. At this point though we
have a problem because 18 lt 24 But 55 gt
30 Therefore, neither column dominates the other.
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Simple games - 2Columns choice cont.
However, when column examines this game, column
knows that row is going to select row 2.
Therefore, columns only real choice is between
paying 55 and paying 30. Column will select col
2, and lose 30 to row in this game. Notice the
you know, I know logic.
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Simple games - 3
Answer the following 3 questions before going to
the following slides. What should row do?
(difficult question) What should column do?
(difficult question) What is the value of the
game (double difficult question since the first
two questions are difficult)
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Simple games - 3
This game has no dominant row nor does it have a
dominant column. Thus, we have no straightforward
answer to this problem.
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Simple games - 3Rows conservative approach
Row could take the following conservative
approach to this problem. Row could look at the
worst that can happen in either row. That is, if
row selects row 1, row may end up winning only
25 whereas if row selects row 2 row may end up
winning only 14. Therefore, row prefers row 1
because the worst case (25) is better than the
worst case (14) for row 2.
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Simple games - 3Maximin
Since 25 is the best of the worst or maximum of
the minima it is called the maximin.
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Simple games - 3Columns conservative way
Column could take a similar conservative
approach. Column could look at the worst that can
happen in either column. That is, if column
selects col 1, column may end up paying as much
as 34 whereas if column selects col 2 column may
end up paying as much as 67. Therefore, column
prefers col 1 because the worst case (34) is
better than the worst case (67) for column 2.
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Simple games - 3Minimax
Since 34 is the best of the worst or minimum of
the maxima for column it is called the minimax.
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Simple games - 3Solution ???
When we put row and columns conservative
approaches together we see that row will play row
1, column will play column 2 and the outcome
(value) of the game will be that column will pay
row 25 (the outcome in row 1, column 1). What is
wrong with this outcome?
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Simple games - 3Solution ???
What is wrong with this outcome? If row knows
that column will select column 1 because column
is conservative then row needs to select row 2
and get 34 instead of 25.
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Simple games - 3Solution ???
However, if column knows that row will select row
2 because row knows that column is conservative
then column needs to select col 2 and pay only
14 instead of 34.
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Simple games - 3Solution ???
However, if row knows that column knows that row
will select row 2 because row knows that column
is conservative and therefore column needs to
select col 2 then row must select row 1 and
collect 67 instead of 14.
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Simple games - 3Solution ???
However, if column knows that row knows that
column knows that row will select row 2 because
row knows that column is conservative and
therefore column needs to select col 2 and that
therefore row must select row 1 then column must
select col 1 and pay 25 instead of 67 and we
are back where we began.
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Simple games - 3Solution ???
The structure of this game is different from the
structure of the first two examples. They each
had only one entry as a solution and in this game
we keep cycling around. There is a lesson for
this game.. .
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Simple games - 3Solution ???
The only way to not let your opponent take
advantage of your choice is to not know what your
choice is yourself. That is, you must select
your strategy randomly. We call this a mixed
strategy.
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Examination of game 1
Minimax
maximin

• Notice that in game 1 (which is trivial to solve)
  we have that
• maximin minimax

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Examination of game 3
Minimax
maximin

• Notice that in game 3 (which is hard to solve) we
  have that
• maximin lt minimax. vhe Value of the game is
  between maximin, minimax

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Repetition of the game

• Consider game 3 above . What would you choose
• if this game were played only once?
• if this game were played many times?

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Mixed strategies

• Row will pick row 1 with probability p and row 2
  with probability (1-p)
• Column will pick col 1 with probability q and col
  2 with probability (1-q)

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Expected values (weighted average) as a function
of p
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Graph of expected value as a function of rows mix
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Solution

• We need to find p to make the expected values
  against both columns equal
• We need to find q to make the expected values
  against both rows equal

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Example - Results
Row should play row 1 32 of the time and row 2
68 of the time. Column should play column 1 85
of the time and column 2 15 of the time. On
average, column will pay row 31.10.
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Example - value of the game

• .32225.67734 31.097 (Col 1 rows mix)
• .32267.67714 31.097 (Col 2 rows mix)
• .85525.14567 31.097 (Row 1 cols mix)
• .85534.14514 31.097 (Row 2 cols mix)

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Expect value computation (continued)

• If row and column each play according to the
  percentages on the outside then each of the four
  cells will occur with probabilities as shown in
  the table

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Expect value computation (continued)

• This leads to an expected value of
• 25.27867.04734.57914.098 31.097

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Solution summary

• If maximinminimax
• there is a saddle point (equilibrium) and each
  player has a pure strategy plays only one
  strategy
• If maximin does not equal minimax
• maximin lt value of game lt minimax
• We find mixed strategies
• We find the (expected) value or weighted average
  of the game

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Zero-sum Game features

• A constant can be added to a zero sum game
  without affecting the optimal strategies.
• A zero sum game can be multiplied by a positive
  constant without affecting the optimal
  strategies.
• A zero sum game is fair if its value is 0
• A graph can be drawn for a player if the player
  has only 2 strategies available.

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Non zero-sum games (Material not in text)

• To this point we have examined zero sum games.
  Our notation for non zero-sum games will be a
  little different.

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Non zero sum gamesNotation

• Payoff listed for each player
• row receives the first payoff
• Column receives the second payoff
• Both players receive (maximize)

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Non zero sum gamesExample

• If row picks row 2 and column picks column 1 then
  row receives 33 and column receives 51. Both want
  to receive as much as possible.

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Special games

• Constant sum game
• a1a2b1b2c1c2d1d2
• Zero sum game
• a1a2b1b2c1c2d1d2 0

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Constant Sum GameExample

• Notice that all cells sum to 100. This is a
  constant sum game that could well represent
  market share in a duopoly.

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Constant Sum GameConversion to Zero-sum

• A constant sum game can be converted to a zero
  sum game by simply dropping the second payoff