MATH 4321 Game Theory(Spring, 2014)
Tutorial Note III
Couse Review
1. Pure strategy
A pure strategy is a players complete plan for playing the game. It should cover every contingency.
A pure strategy for a Player is a rule that tells him exactly what move to make in each of his
information sets. It should specify a particular edge leading out from each information set1 .
2.Reduced pure strategy
Specification of choices at all information sets except those that are eliminated by the previous
moves2 .
3.Random payoffs
The actual outcome of the game for given pure strategies of the players depends on the chance
moves selected, and is therefore a random quantity. We represent random payoffs by their average
values3 .
4.Game in strategic form
• Set of players: {1, · · · , n}
• A set of pure strategies for player i, Xi , i = 1, · · · , n.
• Payoff function for the ith player.
u i : X1 × · · · × X n → R
5.Solution Concept: Strategic Equilibrium
Definition. A vector of pure strategy choices (x1 , x2 , · · · , xn ) with xi ∈ Xi for i = 1, · · · , nis said
to be a pure strategic equilibrium, or PSE for short, if for all i = 1, 2, · · · , n, and for all x ∈ Xi ,
ui (x1 , · · · , xi−1 , xi , xi+1 , · · · , xn ) ≥ ui (x1 , · · · , xi−1 , x, xi+1 , · · · , xn ).
(1)
Equation (1) says that if the players other than player i use their indicated strategies, then the best
player i can do is to use xi . Such a pure strategy choice of player i is called a best response to the
strategy choices of the other players. It is self enforcing!
6.Finding All PSEs. (2 Person)
• Put an asterisk after each of Player Is payoffs that is a maximum of its column.
• Put an asterisk after each of Player IIs payoffs that is a maximum of its row.
• Then any entry of the matrix at which both Is and IIs payoffs have asterisks is a PSE, and
conversely.
7.Method of Backward Induction
Starting from any terminal vertex and trace back to the vertex leading to it. The player at this vertex
will discard those edges with lower payoff. Then, treat this vertex as a terminal vertex and repeat
the process. Then, we get a path from the root to a terminal vertex
Theorem: The path obtained by the method of backward induction defines a PSE.
1
Even for a simple game, there may be a large number of pure strategies.
It is not too easy to find the set of reduced pure strategies. Also we need to use full pure strategies for another
important concept.
3
In representing the random payoffs by their averages, we are making a rather subtle assumption. We are saying
that receiving $5 outright is equivalent to receiving $10 with probability 0.5. The proper setting for this concept is
Utility Theory developed by von Neumann and Morgenstern.
2
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note III
In fact, the PSE obtained by the method of backward induction satisfies stronger properties so
that it is called a perfect pure strategy equilibrium.
Definition: A subgame of a game presented in extensive form is obtained by taking a vertex in
the Kuhn tree and all the edges and paths originated from this vertex.
Definition: A PSE of a game in extensive form is called a Perfect Pure Strategy Equilibrium
(PPSE) if it is a PSE for all subgames.
Theorem: The path obtained by the method of backward induction defines a PPSE.
8. Zero-sum game
For a two-person zero-sum game, the payoff function of Player II is the negative of the payoff of
Player I, so we may restrict attention to the single payoff function of Player I.
Definition: The strategic form, or normal form, of a two- person zero-sum game is given by a
triplet (X, Y, A), where
• X is a nonempty set, the set of strategies of Player I
• Y is a nonempty set, the set of strategies of Player II
• A is a real-valued function defined on X × Y . (Thus, A(x, y) is a real number for every
x ∈ X and every y ∈ Y .)
The interpretation is as follows. Simultaneously, Player I chooses x ∈ X and Player II chooses
y ∈ Y , each unaware of the choice of the other. Then their choices are made known and I wins the
amount A(x, y) from II.
9. Matrix Games
A finite two-person zero-sum game in strategic form, (X, Y, A), is sometimes called a matrix game
because the payoff function A can be represented by a matrix. If X = {x1 , · · · , xm } and Y =
{y1 , · · · , yn }, then by the game matrix or payoff matrix we mean the matrix


a11 · · · a1n

.. 
A =  ...
. 
am1 · · ·
amn
where aij = A(xi , yj ). In this form, Player I chooses a row, Player II chooses a column, and II pays
I the entry in the chosen row and column. Note that the entries of the matrix are the winnings of
the row chooser and losses of the column chooser.
10.Removing Dominated Strategies
Definition. We say the ith row of a matrix A = (aij ) dominates the kth row if aij ≥ akj for all
j. We say the ith row of A strictly dominates the kth row if aij > akj for all j. Similarly, the jth
column of A dominates (strictly dominates) the kth column if aij ≤ aik (resp. aij < aik ) for all i.
Problems
1. Find the pure strategies In this problem, there is a game tree as follow:
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note III
Figure 1: Game tree
a) What are the pure strategies for player 2?
b) What are the pure strategies for player 1?
Solution 1:
a) S2 = {(C, E); (C, F ); (D, E); (D, F )}
b) S1 = {(B, G); (B, H), (A, G), (A, H)}
This is true even though, conditional on taking A, the choice between G and H will never have to
be made.
2. Convert an extensive-form game into normal form
Convert the problem 1 to normal form.
Solution 2:
(C,E)
(A,G) (3,8)
(A,H) (3,8)
(B,G) (5,5)
(B,H) (5,5)
(C,F) (D,E)
(3,8) (8,3)
(3,8) (8,3)
(2,10) (5,5)
(1,0) (5,5)
(D,F)
(8,3)
(8,3)
(2,10)
(1,0)
This illustrates the lack of compactness of the normal form, games aren’t always this small,
even here we write down 16 payoff pairs instead of 5.
3. Find the perfect pure strategy equilibria of problem 1.
Solution 3:
By using the asterisk formula we have
(C,E)
(C,F)
(D,E)
(D,F)
(A,G) (3, 8∗ ) (3∗ , 8∗ ) (8∗ , 3) (8∗ , 3)
(A,H) (3, 8∗ ) (3∗ , 8∗ ) (8∗ , 3) (8∗ , 3)
(B,G) (5∗ , 5) (2, 10∗ ) (5, 5) (2, 10∗ )
(B,H) (5∗ , 5∗ ) (1, 0) (5, 5∗ ) (1, 0)
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note III
Let’s using the game tree backward induction again,
Figure 2: Find the PPSE by backward induction
I am sorry that there is something wrong in previous notes here.
The Matrix form can find the PSE is [(A,G), (C,F)], [(A,H), (C,F)] and [(B,H), (C,E)], but you
should implement the formula in correct columne or row4 .
In sum, the PPSE of this game is [(A,G), (C,F)].
An interesting game
Rock-paper-scissors-lizard-Spock5
Rock-paper-scissors-lizard-Spock is an expansion of the classic selection method game rockpaper-scissors. It operates on the same basic principle, but includes two additional weapons: the
lizard (formed by the hand as a sock-puppet-like mouth) and Spock(formed by the Star Trek Vulcan
salute). This reduces the chances of a round ending in a tie (from 1/3 to 1/5). The game was
invented by Sam Kass with Karen Bryla, as ”Rock Paper Scissors Spock Lizard”.
The game was mentioned in four episodes of The Big Bang Theory. According to an interview with Kass, the series producers did not ask for permission to use the game, but he was later
referenced in an episode in the fifth season for which he thanked them on his website.
The rules of Rock-paper-scissors-lizard-Spock are:
•
•
•
•
•
•
•
•
•
•
4
5
Scissors cut paper
Paper covers rock
Rock crushes lizard
Lizard poisons Spock
Spock smashes scissors
Scissors decapitate lizard
Lizard eats paper
Paper disproves Spock
Spock vaporizes rock
Rock crushes scissors
In exam, I think you should analyse the problem(PSE or PPSE) first and choose the correct method.
cited from wiki
MATH 4321 Game Theory(Spring, 2014)
(a) fig.1
Tutorial Note III
(b) fig.2
Figure 3: Rock-paper-scissors-lizard-Spock
There are ten possible pairings of the five gestures; each gesture beats two of the other gestures and
is beaten by the remaining two. As with the original rock-paper-scissors game, if two players pick
the same gesture, it is a tie. The original rules (rock beats scissors, scissors beats paper, paper beats
rock) still remain the same.
Solution of this game:
Figure 4: The normal form matrix of Rock-paper-scissors-lizard-Spock. Rows represent available
choices for player 1, columns those for player 2. Numbers in cells show utility (payoff) for player
1, player 2.