Lisandro Rodriguez
Proposal Game Theory
04-08-10
Game Theory
Game theory is the study of how players should rationally play games. In many
cases, game theory analyzes situations in which there are at least two players, who may
find themselves in conflict because of different goal and objectives. The outcomes
depend on the choices of all the players. Each player would like the game to end in an
outcome which gives him as large a payoff as possible. The control over the outcome is
depended on the strategy the player uses.
My project is be based on using game theory in solving games with strategies that
leads to outcomes, which describes the consequences of each player’s choice. I will
solve problems with a payoff matrix that the rows and columns corresponding to the
strategies of the two players and the numerical entries give the payoffs to the players
when these strategies are selected. I will approach problems that are Partial-Conflict
Games, which is a variable-sum game in which both players can benefit by cooperation
but may have strong incentives not to cooperate. Total Conflict Games, while one player
wins the other player lose. Total Conflict Games deals with pure strategies and mixed
strategies. Most of these games I will be solving are non-zero sum games and zero sum
games with optimal strategies. Zero sum game is one in which the payoff to one player is
the negative of the corresponding payoff to the other, so the sum of the payoffs to the
players is always zero. Non-Zero sum game describes a situation in which the interacting
parties' aggregate gains and losses is either less than or more than zero. Overall,
throughout my project I cover problems based on Total Conflict Games: pure & mixed
strategy and Partial-Conflict games.
Total Conflict Game: Mixed Strategy
Bob have the choice of either parking illegally on the street or park in the lot and pay
$16. Parking illegally is free if the police officer is not patrolling, but you will receive a
$40 parking ticket if she is. However, Bob peeved when he has to pay to park in the lot
on days when the officer does not patrol, and Bob is willing to assess this outcome as
costing $32 ($16 for parking plus $16 for your time, inconvenience, and grief). It seems
reasonable to assume that the police officer ranks her preferences in the order (1) giving
Bob a ticket, (2) not patrolling when Bob is parked in the lot, (3) patrolling with Bob in
the lot, and (4) not patrolling with Bob parked illegally.
Lisandro Rodriguez
Proposal Game Theory
04-08-10
(a) Describe this as a matrix game, assuming that Bob is playing a zero-sum game
with the officer.
(b) Solve this matrix game for it optimal mixed strategies and its value.
(c) Discuss whether it is reasonable or not to assume that this game is zero – sum.
(A)
Officer does not patrol
You park in street
You park in lot
Officer patrols
0
-40
-32
-16
Zero-sum game is a constant-sum game in which the payoff to one player is the
negative of the payoff to the other player, so the sum of the payoffs to the players at
each outcome is zero. The matrix game shown above is according to the parking
situation where Bob does not lose money when the officer is not in patrol and parks in
the street. Bob does lose money on the other occasions such as when the officer is not
on patrol and he parks in the lot or when the officer patrols and Bob park in the street.
This diagram is the assumption that Bob is playing a zero-sum game with the officer.
Evidently, Bob is assuming to play a zero-sum game. This diagram represents Bob
point of view so everything Bob looses the officer will gain, if the diagram is seen in
the officer point of view.
(B) Optimal mixed strategy guarantees that the resulting payoff is the best that this
player can obtain against all possible strategy choices by an opponent. By expressing
a mixed strategy we need the equation (pBob, pOfficer) = (p1, p2) = (p, 1-p).
Officer does not patrol
(ONP)
Bob park in the street (BPS)
Bob parks in the lot (BPL)
Officer patrols
(OP)
0
-40
-32
-16
To express this mixed strategy equation we need to form a linear equation where we
can find both Bob: (pBob, pOfficer) and Officer: (pOfficer, pBob) so the optimal mixed
strategy for Bob is
Lisandro Rodriguez
Proposal Game Theory
04-08-10
(ONP) = (0)p + (-32)(1-p) = 32p-32
(OP) = (-40)p + (-16)(1-p) = -24p-16
Solving for p
32p-32=-24p-16 → p=2/7
Now that we found p which was pBob , then (p, 1-p) = (2/7,(1-2/7)) =
Bob: (2/7,5/7).
Further, to find the optimal mixed strategy of the Officer: (pOfficer, pBob) we need to
calculate a linear equation but instead of taking the numbers vertically as we did in
Bob case, we are going to use the numbers horizontally.
(BPS) = (0)p + (-40)(1-p) = 40p – 40
(BPL) = (-32)p + (-16)(1-p) = -16p – 16
Solving for p
40p – 40 = -16p – 16 → p= 3/7
To solve for , pBob =(1-p) = (1-3/7) pBob= 4/7
However, since in Bob case we had that the pOfficer = 5/7, then to calculate pBob in
the officer case we need to subtract (5/7)-(4/7) which will give us 1/7. In addition,
the officer’s optimal mixed strategy is Officer: (1/7,4/7)
To calculate the value of the optimal mixed strategy we take the equation
p + (1-p) in which we used to find the optimal mixed strategy and plug in the p
values that was solve above. For instance,
(ONP) = (0)p + (-32)(1-p) = 32p-32 and (OP) = (-40)p + (-16)(1-p) = -24p-16
optimal strategy were Bob: (2/7,5/7). So, if we plug in the p values for any linear
equation we solved, we can find the values of the optimal mixed strategy.
Bob: (pBob, pOfficer)
Bob: (2/7,5/7)
(0) p + (-32)(1-, p) =
(0)2/7 + (-32)(1-2/7) = -22.86
(C) The diagram that I created was design in Bob points of view where his payoff was
his lost except for him parking in the street while the officer was not on patrol. It was
design to see how Bob was affect if he parked on the street or on the parking lot while
Lisandro Rodriguez
Proposal Game Theory
04-08-10
the officer was on patrol or not. Further, since Bob looses money toward the officer
the payoff of the officer’s are no different than Bob. The officer will always benefit
when Bob do not making it assumable to be a zero-sum game.
Partial-Conflict Game
Discuss the player’s possible behavior when these games are played non-cooperative
manner (with no prior communication or agreements). The first payoff is to the row
player; the second, to the column player. Is the Nash equilibrium in this game sensible?
Why or why not?
Player II
Cooperative
Cooperative
Non-Cooperative
(4,4)
(1,3)
(3,1)
(2,2)
Player I
Non Cooperative
Non-cooperative games are games in which a binding agreement cannot be
enforced. Even if communication is allowed in such games, there is no assurance that a
player can trust an opponent to choose a particular strategy that he or she promise. For
instance, there are two players who committed a crime and not enough evident is
determine in the crime scene. The FBI agents need to make them confess that they did
the crime.
Player I and Player II are guilty of a bank robbery that occurred during the
recession. However, the FBI agents do not have enough evident to put them away for the
time they deserve. The FBI agents took both Player I and Player II to their headquarter
and into separate rooms where Player I and Player II were interrogated. During the
interrogation, the FBI agents told both Players (in their separate rooms) that they had a
choice to cooperate (answering all questions and rat out their partner) or not to
cooperate (stay quiet and giving the agents a hard time). The FBI agents also mentioned,
“If both you guys cooperate you will only spend 4 years in prison. However if one of you
guys cooperate and the other one does not, then the guy who cooperate will have 1 year
Lisandro Rodriguez
Proposal Game Theory
04-08-10
in prison and the non-cooperative will spend 3 years in prison. Now, if both don’t
cooperate, then you both will go to jail for 2 years.” Both players have no prior
communication or agreement because they are in separate rooms. Player I want to
assume that his partner is a noble person and will not cooperate in which will allow
Player I to cooperate and receive less time in prison. In essence, Player II is assuming the
same idea as Player I where Player II will cooperate and only suffer with 1 year in prison
while Player I suffers with 3 years in prison. Further, if both players cooperate because
they want to assume that one of then won’t cooperate, then both players will suffer the
maximum time in jail. However, both players want less time in prison but feel that they
cannot betray each other by cooperating. How many years will both players decide to
receive?
Nash equilibrium is strategies associated with an outcome such that no player can
benefit by choosing a different strategy, given that the other players do not depart from
their strategies. Both (cooperate, cooperate) and (non-cooperate, non-cooperate) on the
top diagram are Nash equilibrium because when every player in a game plays his
dominant strategy the player earns a larger or better payoff than any other (assuming
every player has a dominant strategy) which is always better than any other strategy
depending on the situation, then the outcome will be a Nash equilibrium.
Total Conflict Game: Pure Strategy
(a) Describe in detail one pure strategy for the player who moves first in tic-tac-toe.
(This strategy must tell how to respond to all possible moves of the other player.)
(b) Is your strategy optimal in the sense that it will guarantee the first player a tie (and
possibly a win) in the game?
(A) Tic-tac-toe is a game with two players alternately placing an X or an O,
respectively, in one of the nine unoccupied spaces in a 3X3 grid. The winner is the first
player to have three X’s, or three O’s, in either the same row, the same column, or along
a diagonal. Part A wants a solution that describes one pure strategy for the player who
moves first in tic-tac-toe. A pure strategy significantly means that player does not
involve randomized choices. So, if Sam is X and Bob is O, then let Sam mark X first in
the center box or the grid (Sam is starting a strategy by placing an X in the center, note
X
O
O
Lisandro
X Rodriguez
X
X
Proposal Game Theory
04-08-10
X
X
X
X X O X X O X X
O O X O O X O O X O O X O O X O O
X
X
O X
O X
O X X O X
that if Sam places an X in any unoccupied space other than the center, it will require Sam
to change his strategy). If Bob moves next to a corner box or to a side box, (Bob is
thinking of strategies to counter Sam from winning, and a strategy to win the game.) Sam
moves to a corner box in the same row or column. It becomes anyone’s game after Sam
second move. There are now six more boxes to fill, and Sam have up to three more
moves (if Sam or Bob does not win before this point, then Sam and Bob will have a tie),
but the rest of Sam strategy becomes quite complicated, involving moves like blocking
the completion of a row/column/diagonal by Bob. In addition, Sam had a reason and a
strategy for every X he placed on the grid. There was never any random move from Sam
or Bob, which determine the definition of a pure strategy. In the diagram below, you can
see that it gets complicated for the six blank spaces because Sam can win, counter, or lose
the game.
(B) Optimal strategy guarantees that the resulting payoff is the best that a player
can obtain against all possible strategy choices by an opponent. So in Sam’s case, he
must show that he is guarantee at least a tie, no matter what choices Bob makes.
Assuming Sam is X and Bob is O let Sam place an X first in any corner box of the grid.
If Bob places an O in the center of the grid, then let Sam place an X in the corner box of
the same row or column, respect to his first move. Now, Bob needs to counter Sam’s
move in order to stop Sam from winning. After this move, Bob can place an O in any
unoccupied space and it will lead to a tie. The next move, Sam needs to counter Bob
from winning. Once Sam counters Bob, Bob attempts to win and place an O in the side
of the grid (not the corner). Moreover, the game is lead to a tie because Bob and Sam are
countering each other from winning. In the diagram, the countering started in the fourth
and fifth move, which made it difficult for Sam to regain the offensive side because Bob
took over the game. After Bob was on the offensive side, Sam had to counter every
move Bob did. Overall, Bob and Sam optimal strategy lead them to a tie in tic-tac-toe.
X
O
O
Lisandro
X Rodriguez
X
X
Proposal Game Theory
04-08-10
X
X
X
X O X X O
O
O
O
O O
O O X O O X O O X
X
X X O X X O X X O X X O X X O X X O X
O
X
X
X
X X
X X O X X O X X O X X O X X O X X O X
O
O
O
O O X O O X O O X O O
X
X
X
O X
O X X
X
X
X
X O
X O
X O
X O X
O O X O O X O O X O O X O O X O O X
X
X
X
X X
X X O X X O
O
X
X
X
X
X X O X X O X X
O O X O O X O O X O O X O O X O O
X
X
O X
O X
O X X O X
X
O
O
X
Bob is
Sam is
countering
countering
Sam
started
strategy
toe.
Bob
started
strategy
Sam
going
for the
win
These are
situation where Sam
both lead to a tie in
Bob
counters
Bob is
trying
to go
for the
win
Sam is
trying
to go
for the
win
different
and Bob
tic-tac-
Tie!
Lisandro Rodriguez
Proposal Game Theory
04-08-10
Note: all of my example problems came from the book, For All Practical
Purposes by W.H. Freema.
Reference
For All Practical Purposes by W.H. Freeman
Game Theory and Srategy by Philip D. Straffin