MT369, Game Theory Part 1, diagrams etc.
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Basic definitions.
Game: A set of rules for interaction between a number of players. Starting with an initial state
(defined uniquely by the rules) each player in turn makes a move , choosing one of several options
determined by the rules.
A play of a game is a sequence of moves starting from the initial state and conforming to the rules of
the game.
State of a game: The current position of a game together with the complete set of moves that led to
this position.
Final state: one in which no further moves are allowed. For each final state of a game, the rules
normally specify a payoff to each player.
Length of a game: The maximum number of moves in any play of the game (if this is finite).
Finite game: one with only finitely many states. Equivalently: a game of finite length in which the
player to move has only finitely many legal moves.
Extensive form of a game: Diagram showing one node for every state of the game; at each non-final
node (P , say) a set of arrows pointing to every state that can immediately follow P ; and for every
possible state, some indication of all the other states in the same information set.
Example 1.1* : 2—2 Nim. Rules: there are 2 players called A , B who move alternately; A
moves first. Initial position: 2 piles of 2 matches. Moves: At each turn, the player with the move
must take any number of matches ( > 0 ) from one pile only. Whoever removes the last match loses†.
2 2
A 1
Extended tree: Notation: The
letter in each box is player to
move. Numbers in large type are
the sizes of the piles, and each
1 2
B 2
state is numbered in small type
for later reference.
0 2
A 4
0 1
B 9
Winner:
0 0
10
1 1
A 5
1 0
A 6
0 1
A 7
1 0
B 11
0 0
0 0
12
13
0 0
0 0
14
15
A
B
0 2
B 3
A
B
B
0 0
8
A
*
These numbers refer to A.J.Jones's book. I have rearranged things, but kept the numbers.
†
This is what the book says and I am going along with it. But I believe that normally whoever removes the last
match wins
MT369, Game Theory Part 1, diagrams etc.
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Pure Strategy: A prior, comprehensive decision by one player of the choices to be made at every
decision point that that player might possibly meet.
Pure Strategies for A
Table 1.1. Pure strategies for
2—2 Nim. In this table the
expression x  y means "If I
find myself at box x then move
to y" .
Pure Strategies for B
Name
Moves
Name
Moves
A1
12 , 49
B1
24 , 37
B2
25 , 37
B3
26 , 37
B4
24 , 38
B5
25 , 38
B6
26 , 38
A2
12 , 410
If A chooses A3 then the game
can never reach state 4 so there
is no entry 4x .
A3
13
Definition. The normal form of a game can only be constructed for a 2 person game. It is a matrix in
which the i-j-th cell shows the outcome if the first player chooses hir i-th pure strategy and the 2nd
player chooses hir j-th strategy. Usually the entry in each cell shows the final state and some
indication of the payoffs to the players. In matrix theory, rows always come before columns. So in the
normal form the rows of the matrix correspond to the first player's strategies, and the columns
correspond to the 2nd player's strategies.
Table 1.4. Normal form of 2—2 Nim. Each cell gives the final state & name of winner.
This table contains two deliberate mistakes, Ex 1 Q 1 is to find these.
B1
B2
B3
B4
B5
B6
A1
A(14)
A(15)
B(12)
A(14)
A(15)
B(12)
A2
B(10)
A(15)
B(13)
B(12)
A(15)
B(12)
A3
B(13)
B(13)
B(13)
A(8)
A(8)
A(8)
Definition. A game is zero sum if the sum of the payoffs to all players is always 0 . If we are
considering a 2-person zero sum game, then its outcome in any final state can always be represented
by a single number, which is the payoff to the first player. If this is £x then we know that the payoff
to the 2nd player is £x so we do not need to record it.
Definition. The matrix form of a 2 person zero
sum game is a matrix in which the i-j-th cell
gives the payoff to Player 1 if (s)he adopts hir
i-th strategy and Player 2 adopts hir j-th
strategy.
B1
B2
B3
B4
B5
B6
A1
1
1
-1
1
1
-1
A2
-1
1
-1
-1
1
-1
A3
-1
-1
-1
1
1
1
Table 1.5. Matrix form of 2—2 Nim,
assuming that the loser pays the winner £1 . With this payoff rule the game is zero sum. Each cell
gives the payoff to the first player A (whose strategies are in the rows).
MT369, Game Theory Part 1, diagrams etc.
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If a 2 person game is not zero sum, we can construct a bimatrix form of the game. The entry in cell
i-j has the form (p,q) where p is the payoff to Player 1 and q ditto 2
Example: "Anti-inflationary" 2—2 Nim . The game is the same but with different payoffs. Loser
must pay winner £1 and also burn‡ £n where n is the number of moves. In order to construct the
bimatrix of the game it is a good idea to build an intermediate table ( 1.7 ). This gives the payoffs to
A and B from every terminal state of the game.
State
14
10
15
12
13
8
Payoff to A
1
-4
1
-4
-4
1
Payoff to B
-5
1
-5
1
1
-3
Then you can combine this with the normal form (Table 1.5) to get the Bimatrix form of the game
"Anti-inflationary" 2—2 Nim. Each cell gives the payoffs to A and B in that order.
B1
B2
B3
B4
B5
B6
A1
(1,-5)
(1,-5)
(-4,1)
(1,-5)
(1,-5)
(-4,1)
A2
(-4,1)
(1,-5)
(-4,1)
(-4,1)
(1,-5)
(-4,1)
A3
(-4,1)
(-4,1)
(-4,1)
(1,-3)
(1,-3)
(1,-3)
Definitions: A game has perfect information if the player to move always knows what state the game
is in. When a game does not have perfect information, an information set is the set of all states that
the game might be in, consistent with the information that is available to the player who is about to
move.
Theorem (proof later) In any finite non-cooperative game of perfect information, each player has a
definite optimum strategy that maximises hir
payoff.
Example 1.2. Renee v Peter. Played with 3
cards: King, Ten and Two. First: R
chooses one of these cards & puts it face
down on the table. Then P must call High
or Low and the card is shown. If P called
correctly ( King = high, two = low ) he
wins £3 from R ; if he is wrong he loses
£2 . If R's card was the 10 then P wins £2
if he called Low. But if P called High ,
then R must choose a 2nd card and P must
call again. This time, he wins £1 if he calls right and loses £3 if he called wrong.
Here the numbers at each final state are the payoffs to the first player (Renee).
‡
When AJJ's book was written, pounds were paper.
MT369, Game Theory Part 1, diagrams etc.
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Table 1.2. Pure strategies for R & P .
Table 1.9
Renee
Peter
Matrix Form
P1
P2
P3
R1. K
P1. High, then High
R1
-3
-3
2
R2. Ten then K
P2. High, then Low
R2
-1
3
-2
R3. Ten then Two
P3. Low
R3
3
-1
-2
R4
2
2
-3
Renee
R4. Two
Peter
In this table 1.2, " X then Y " means that you will play Y if you get the opportunity.
If a game involves any events that are due to chance, all chance events are supposed to be chosen by an
extra player called Nature. Nature chooses all her moves randomly, according to probabilities which
are fixed at any state of the game.
Definition. Given some random event that has a definite set of possible outcomes, we say that a
particular outcome (  , say) has probability p if we believe that this outcome will occur approximately Mp times in a sequence of M trials. All probabilities are real numbers in 0  p  1 . We
write p() for the probability of  . So for example when we toss a die, we usually assume that all 6
values are equally probable . So each value has probability 1/6 .
Example 1.3. Red—Black . A is dealt a card. If
it is red, then A can either pass the move to B or
call for a coin to be tossed. If the card is black, A
must pass the move to B . If B gets the move, he
must guess Red or Black. The payoffs are as
shown. Note that the first move is by Nature .
When we draw the extensive form of a game that
involves chance, we label all of Nature's possible
moves with their probabilities.
N
R (½)
B (½)
A
A
N
B
H (½)
T (½)
p
q
R
r
Table 1.3. List of pure strategies for Red—Black.
A
B
Pass
Pass
Toss
Nature
A1:
Toss
B1:
Call R
N1:
Red, then Heads (¼)
A2:
Pass
B2:
Call B
N2:
Red, then Tails (¼)
N3:
Black (½)
R
B
s
t
B
u
MT369, Game Theory Part 1, diagrams etc.
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Table 1.10. Intermediate payoff table for Red—Black.
N1 (¼)
N2 (¼)
N3 (½)
B1
B2
B1
B2
B1
B2
A1
p
p
q
q
t
u
A2
r
s
r
s
t
u
Definition. Let x be a random variable that is a number determined by Nature. The mean of x or
the expected value of x written E(x) is the sum:
E(x) =  p()
summed over all possible values of x where p() stands for the probability that x has the value  .
Equivalently: if we ask Nature to choose a large number of values of x then we expect that the
arithmetic mean of these values will be "close" to E(x) .
Table 1.11. Matrix form of Red—Black . This shows the
mean value of the payoffs to A assuming that they choose their
respective strategies and that Nature behaves herself.
B1
B2
A1
(p+q+2t)/4
(p+q+2u)/4
A2
(r+t)/2
(s+u)/2
Definition. A game is non-cooperative if the players are unable to make any sort of bargains about
how they will play. So each player's payoff is determined by the rules of the game, for every terminal
state. Owing to limitations of time we can only consider non-cooperative games in this course. Also
we will assume that each player is trying to maximise hir payoff. Also we assume that all the players
are intelligent . An intelligent player is one who can find the entire game tree, list all available
strategies for all players, and thus find the outcome of any set of strategies chosen by all players,
including Nature.
In the next section we will consider only 2-person zero-sum games between intelligent players. From
the above analysis we can see that any such game is equivalent to the following game:
Definition. A matrix game is played by 2 players with a fixed matrix M . A secretly chooses a row,
(say, row i ) and B chooses a column (say, j ). Then they reveal their choices and B pays A the
number in Mij .