3.2.1. Static Games of complete information: Dominant
Strategies and Nash Equilibrium in pure and mixed
strategies
- Some Games -
Strategic Behavior in Business and Econ
Outline
3.1. What is a Game ?
3.1.1. The elements of a Game
3.1.2 The Rules of the Game: Example
3.1.3. Examples of Game Situations
3.1.4 Types of Games
3.2. Solution Concepts
3.2.1. Static Games of complete information: Dominant
Strategies and Nash Equilibrium in pure and mixed
strategies
3.2.2. Dynamic Games of complete information: Backward
Induction and Subgame perfection
Strategic Behavior in Business and Econ
Reminder
Solution concepts for this type of games
Equilibrium in Dominant Strategies
When there is an “always winning” strategy
Equilibrium by elimination of Dominated Strategies
When there are “worse than” strategies
Nash Equilibrium
Works in any case
In pure strategies (players do not randomize)
In mixed strategies (players do randomize)
Strategic Behavior in Business and Econ
Reminder
If you have a Dominant Strategy use it, and expect your
opponent to use it as well
If you have Dominated Strategies do not use any of them,
and expect you opponent not to use them as well
(eliminate them from the analysis of the game)
If there are neither Dominant Strategies nor Dominated
Strategies, look for Nash Equilibria and play accordingly.
Expect your opponent to play according to the Nash
Equilibrium as well
If there are no Nash Equilibria in pure strategies, play at
random (mixed strategy) to keep your opponent guessing
When playing randomly, do not follow any pattern, and try
to discover patterns in your opponent's behavior
Strategic Behavior in Business and Econ
Some games
Hard to play games
Game of Chicken (original)
Cold War
Hawk-Dove Game
Hard to solve games
Stag hunt Game
Volunteer's Dilemma
Hard to believe games
Traveler's Dilemma
Guess 2/3 of the average game
Hard to analyze games
Hotelling Spatial Competition Game
Strategic Behavior in Business and Econ
Hard to play games
These are games that, although having a solution, they are
difficult to play if encountered in real life for they correspond
to extreme situations
Game of Chicken (original)
Cold War
Hawk-Dove Game
Strategic Behavior in Business and Econ
Game of Chicken (original)
The game of Chicken models two drivers, both headed for a single
lane bridge from opposite directions. The first to swerve away
yields the bridge to the other. If neither player swerves, the result is
a fatal head-on collision. It is presumed that the best thing for each
driver is to stay straight while the other swerves (since the other is
the "chicken" while a crash is avoided). Additionally, a crash is
presumed to be the worst outcome for both players. This yields a
situation where each player, in attempting to secure his best
outcome, risks the worst.
Strategic Behavior in Business and Econ
Game of Chicken (original)
Strategic Behavior in Business and Econ
The environment of the game
Players:
Strategies:
Payoffs:
Driver 1 and Driver 2
To swerve or To stay straight
(to be defined)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Conflict (anti-coordination)
Symmetric
Strategic Behavior in Business and Econ
The payoffs
This is one of those games in which there is no monetary
payoff. Thus, we must define the payoffs in accordance to
the nature of the game. For each player, we have that:
Best outcome
Second Best
Third Best
Worst Outcome
Strategic Behavior in Business and Econ
The payoffs
This is one of those games in which there is no monetary
payoff. Thus, we must define the payoffs in accordance to
the nature of the game. For each player, we have that:
Best outcome
Second Best
Third Best
Worst Outcome
I stay straight while the other swerves (a)
Strategic Behavior in Business and Econ
The payoffs
This is one of those games in which there is no monetary
payoff. Thus, we must define the payoffs in accordance to
the nature of the game. For each player, we have that:
Best outcome
Second Best
Third Best
Worst Outcome
I stay straight while the other swerves (a)
We both swerve (b)
Strategic Behavior in Business and Econ
The payoffs
This is one of those games in which there is no monetary
payoff. Thus, we must define the payoffs in accordance to
the nature of the game. For each player, we have that:
Best outcome
Second Best
Third Best
Worst Outcome
I stay straight while the other swerves (a)
We both swerve (b)
I swerve and the other stays straight (c)
Strategic Behavior in Business and Econ
The payoffs
This is one of those games in which there is no monetary
payoff. Thus, we must define the payoffs in accordance to
the nature of the game. For each player, we have that:
Best outcome
Second Best
Third Best
Worst Outcome
I stay straight while the other swerves (a)
We both swerve (b)
I swerve and the other stays straight (c)
We both stay straight (crash !) (d)
Where a > b > c > d
Strategic Behavior in Business and Econ
The Game of Chicken (original)
Driver 2
Swerve
Stay Straight
Swerve
b, b
c,a
Stay
Straight
a,c
d, d
Driver 1
Where a > b > c > d
Strategic Behavior in Business and Econ
The Game of Chicken (original)
These games, were the
value of the payoff doesn't
matter (only the order)
are called Ordinal Games
Driver 2
Swerve
Stay Straight
Swerve
b, b
c,a
Stay
Straight
a,c
d, d
Driver 1
Where a > b > c > d
Strategic Behavior in Business and Econ
The Game of Chicken (original)
We could give any value to
a, b, c, and d, as long as
they keep the order
a>b>c>d
Driver 2
Swerve
Stay Straight
Swerve
b, b
c,a
Stay
Straight
a,c
d, d
Driver 1
Where a > b > c > d
Strategic Behavior in Business and Econ
The Game of Chicken (original)
Driver 2
Thus, we can find the
“best replies” as usual
Swerve
Stay Straight
Swerve
b, b
c,a
Stay
Straight
a,c
d, d
Driver 1
Where a > b > c > d
Strategic Behavior in Business and Econ
In any solution, one of the drivers must play chicken
Who is the chicken ?
On the one side both player prefer the other to swerve
On the other side the cost of a crash is so high compared
to the cost of being the chicken that it might make sense
to swerve
It is mutually beneficial for the players to play different
strategies (anti-coordination)
Random Strategies might make sense here, but are not
possible to compute (the probabilities depend on the values
of the payoffs, and these values are arbitrary here)
Strategic Behavior in Business and Econ
Bertrand Russell saw in chicken a metaphor for the nuclear
stalemate. His 1959 book, Common Sense and Nuclear Warfare,
not only describes the game but offers mordant comments on
those who play the geopolitical version of it.
‘Since the nuclear stalemate became apparent, the Governments
of East and West have adopted the policy which Mr. Dulles calls
"brinkmanship." This is a policy adapted from a sport which, I am told,
is practiced by some youthful degenerates. This sport is called
"Chicken!" It is played by choosing a long straight road with a
white line down the middle and starting two very fast cars towards
each other from opposite ends. Each car is expected to keep the
wheels of one side on the white line. As they approach each other,
mutual destruction becomes more and more imminent. If one of
them swerves from the white line before the other, the other, as
he passes, shouts "Chicken!" and the one who has swerved becomes
an object of contempt....’
Strategic Behavior in Business and Econ
‘As played by irresponsible boys, this game is considered decadent and
immoral, though only the lives of the players are risked. But when
the game is played by eminent statesmen, who risk not only their
own lives but those of many hundreds of millions of human beings,
it is thought on both sides that the statesmen on one side are
displaying a high degree of wisdom and courage, and only the statesmen
on the other side are reprehensible. This, of course, is absurd. Both
are to blame for playing such an incredibly dangerous game. The game
may be played without misfortune a few times, but sooner or later
it will come to be felt that loss of face is more dreadful than nuclear
annihilation.
The moment will come when neither side can face the derisive cry of
"Chicken!" from the other side. When that moment is come, the
statesmen of both sides will plunge the world into destruction.’
(From William Poundstone's “The Prisoner's Dilemma”)
Strategic Behavior in Business and Econ
Cold War (based on “Game theory and the Cuban missile crisis”
by Steven J. Brams)
This is a real life (historical) version of a Game of Chicken.
It goes back to the 60's, when the USA and the USSR were
engaged in a so called “cold war”
One of the episodes of the “cold war” was the Cuban missile crisis
The Cuban missile crisis was precipitated by a Soviet attempt in October
1962 to install medium-range and intermediate-range nuclear-armed
ballistic missiles in Cuba that were capable of hitting a large portion of
the United States. The goal of the United States was immediate removal
of the Soviet missiles, and U.S. policy makers seriously considered two
strategies to achieve this end
Strategic Behavior in Business and Econ
A naval blockade (B), or "quarantine" as it was euphemistically
called, to prevent shipment of more missiles, possibly followed
by stronger action to induce the Soviet Union to withdraw
the missiles already installed.
A "surgical" air strike (A) to wipe out the missiles already
installed, insofar as possible, perhaps followed by an invasion
of the island.
The alternatives open to Soviet policy makers were:
Withdrawal (W) of their missiles.
Maintenance (M) of their missiles.
Strategic Behavior in Business and Econ
The environment of the game
Players:
Strategies:
Payoffs:
USA and USSR
For USA: Blockade (B) or Air Strike (A)
For USSR: Withdrawal (W) or Maintenance (M)
(to be defined)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Conflict (anti-coordination)
Symmetric
Strategic Behavior in Business and Econ
The payoffs
This is “Game of Chicken”. Both would like the other to
“swerve”. If they “stay straight”, the crash is the Nuclear War
Best outcome
Second Best
Third Best
Worst Outcome
USA
(A,W)
(B,W)
(B,M)
(A,M)
Strategic Behavior in Business and Econ
USSR
(B,M)
(B,W)
(A,W)
(A,M)
4
3
2
1
Cuban Missile crisis
USSR
The solution(s) is as
before
W
B
USA
M
Compromise
USSR wins
3, 3
2,4
USA wins
Nuclear War
4,2
1, 1
A
Strategic Behavior in Business and Econ
The strategy choices, probable outcomes, and associated payoffs
are a strong simplification of the crisis as it developed over
a period of thirteen days.
Both sides considered more than the two alternatives listed,
as well as several variations on each. The Soviets, for example,
demanded withdrawal of American missiles from Turkey as a quid
pro quo for withdrawal of their own missiles from Cuba, a
demand publicly ignored by the United States.
Most observers of this crisis believed that the two superpowers
were on a collision course
They also agree that neither side was eager to take any
“irreversible step” (more on this next)
Strategic Behavior in Business and Econ
Although in one sense the United States "won" by getting the
Soviets to withdraw their missiles, Premier Nikita Khrushchev of
the Soviet Union at the same time extracted from President
Kennedy a promise not to invade Cuba, which seems to indicate
that the eventual outcome was a compromise of sorts
That is not the prediction of Game Theory !
Negotiation is the key to scape from a mutual bad outcome !
Strategic Behavior in Business and Econ
“Irreversible steps”
One way to scape from a “game of chicken” is to take an
“irreversible step”
What would happen if one of the two drivers (say Driver 1)
defiantly rips off the steering wheel in full view of the other
driver ?
Strategic Behavior in Business and Econ
Driver 2
The driver just eliminated
the option of swerving !
Swerve
Stay Straight
Swerve
b, b
c,a
Stay
Straight
a,c
d, d
Driver 1
Strategic Behavior in Business and Econ
Driver 2
Driver 1 just eliminated
the option of swerving !
Swerve
Stay Straight
Swerve
b, b
c,a
Stay
Straight
a,c
d, d
Driver 1
Strategic Behavior in Business and Econ
Now, there is only one
rational choice for
Driver 2
Driver 2
Swerve
Stay Straight
Driver 1
Stay
Straight
a,c
Strategic Behavior in Business and Econ
d, d
And the game ends with
the preferred outcome
for Driver 1 !!!!!
Driver 2
Swerve
Driver 1
Stay
Straight
a,c
Strategic Behavior in Business and Econ
Sometimes it makes sense to “dispose” of some strategies
But it is very important that your opponent realizes you have done so !
In Stanley Kubrick's Dr. Strangelove. the Russians sought to deter
American attack by building a "doomsday machine," a device that would
trigger world annihilation if Russia was hit by nuclear weapons.
However, the Russians failed to signal. They deployed their doomsday
machine covertly !!
Strategic Behavior in Business and Econ
Hawk-Dove Game
This is version of a Game of Chicken that is very useful in
evolutionary biology
The name "Hawk-Dove" refers to a situation in which two animals
compete for a shared resource and the contestants can choose either
conciliation or conflict.
V is the value of the contested resource, and C is the cost of an escalated
fight. It is (almost always) assumed that the value of the resource is less
than the cost of a fight is, i.e., C > V > 0
If the two animals behave in the same way, the split the resource.
Otherwise, the animal playing Hawk gets the whole resource
Strategic Behavior in Business and Econ
The environment of the game
Players:
Strategies:
Payoffs:
Animal 1 and Animal 2
Dove (show you intention) or Hawk (attack)
(see the table)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Conflict (anti-coordination)
Symmetric
Strategic Behavior in Business and Econ
Animal 2
Dove
Hawk
Dove
V/2 , V/2
0,V
Hawk
V,0
(V-C)/2 , (V-C)/2
Animal 1
Strategic Behavior in Business and Econ
Animal 2
Dove
Hawk
Dove
V/2 , V/2
0,V
Hawk
V,0
(V-C)/2 , (V-C)/2
Animal 1
Notice, since V<C, V > V/2 > 0 > (V-C)/2 It's a Game of Chicken
Strategic Behavior in Business and Econ
Animal 2
The solution(s) is as
before
Dove
Hawk
Dove
V/2 , V/2
0,V
Hawk
V,0
(V-C)/2 , (V-C)/2
Animal 1
Notice, since V<C, V > V/2 > 0 > (V-C)/2 It's a Game of Chicken
Strategic Behavior in Business and Econ
In this case, doesn't seem to be any way to take an
“irreversible step”
In real life, some animals behave as doves while others
Are Hawks (nature plays mixed strategies !)
This example set the basis for a extremely fruitful
application of Game Theory to Evolutionary Biology
(John Maynard-Smith)
And vice versa, Evolutionary Theory can be applied to
Game Theory !
Strategic Behavior in Business and Econ
Hard to solve games
These are social games, situations in which although players seek
a common goal, their individualistic behavior leads to non desirable
outcomes
Stag hunt Game
The Volunteer's Dilemma
Strategic Behavior in Business and Econ
Stag hunt Game
This is a game which describes a conflict between safety and social
cooperation.
Jean-Jacques Rousseau described a situation in which two
individuals go out on a hunt. Each can individually choose to hunt a
stag or hunt a hare. Each player must choose an action without
knowing the choice of the other. If an individual hunts a stag, he
must have the cooperation of his partner in order to succeed. An
individual can get a hare by himself, but a hare is worth less than a
stag. This is taken to be an important analogy for social
cooperation.
Strategic Behavior in Business and Econ
The environment of the game
Players:
Strategies:
Payoffs:
Hunter 1 and Hunter 2
Hunt Stag or Hunt Hare
(to be defined)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Cooperation
Symmetric
Strategic Behavior in Business and Econ
The payoffs
Again, this is a game with no monetary payoff, other than the
value of a Stag or a Hare. We can define the payoffs in
accordance to the nature of the game. For each player, we have
that:
Best outcome
Second Best
Worst Outcome
Strategic Behavior in Business and Econ
The payoffs
Again, this is a game with no monetary payoff, other than the
value of a Stag or a Hare. We can define the payoffs in
accordance to the nature of the game. For each player, we have
that:
Best outcome
Second Best
Worst Outcome
Both hunt Stag
Strategic Behavior in Business and Econ
(a)
The payoffs
Again, this is a game with no monetary payoff, other than the
value of a Stag or a Hare. We can define the payoffs in
accordance to the nature of the game. For each player, we have
that:
Best outcome
Second Best
Worst Outcome
Both hunt Stag
I hunt Hare
Strategic Behavior in Business and Econ
(a)
(b)
The payoffs
Again, this is a game with no monetary payoff, other than the
value of a Stag or a Hare. We can define the payoffs in
accordance to the nature of the game. For each player, we have
that:
Best outcome
Second Best
Worst Outcome
Both hunt Stag
I hunt Hare
I hunt Stag while the other hunts hare
Strategic Behavior in Business and Econ
(a)
(b)
(c)
Stag hunt Game
Hunter 2
Stag
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Stag hunt Game
We could give any value to
a, b, and c, as long as
they keep the order
a>b>c
Stag
Hunter 2
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Stag hunt Game
Hunter 2
We can find the
“best replies” as usual
Stag
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Stag hunt Game
There are 2 equilibria,
but one is better for
the two players than
the other
Stag
Hunter 2
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Stag hunt Game
The equilibrium at which
both player go for the
Stag is called
Payoff Dominant
Stag
Hunter 2
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Stag hunt Game
It corresponds to the
social cooperation
solution
Stag
Hunter 2
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Stag hunt Game
The equilibrium at which
both player go for the
Hare is called
Risk Dominant
Stag
Hunter 2
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Stag hunt Game
It corresponds to the
individually safe
solution
Stag
Hunter 2
Stag
Hare
a, a
c,b
b,c
b, b
Hunter 1
Hare
Where a > b > c
Strategic Behavior in Business and Econ
Individually safe solution
Suppose that Hunter 1 is unsure about what Hunter 2 is going
to do. Suppose she thinks that Hunter 2 is going to go for
Stag with probability p
Hunter 2
Stag (p)
Stag
Hare (1-p)
a, a
c,b
b,c
b, b
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Then, she (Hunter 1) must evaluate the Expected payoff of
each of hear choices: E(Stag) vs. E(Hare)
Hunter 2
Stag (p)
Stag
Hare (1-p)
a, a
c,b
b,c
b, b
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
E(Stag) = p x a + (1-p) x c = c + p (a - c)
E(Hare) = b
Hunter 2
Stag (p)
Stag
Hare (1-p)
a, a
c,b
b,c
b, b
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Hence, Hunter 1 will choose Hare if E(Stag) < E(Hare), that is,
if c + p(a – c) < b
Hunter 2
Stag (p)
Stag
Hare (1-p)
a, a
c,b
b,c
b, b
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Solving for p: c + p(a – c) < b
p(a – c) < (b – c)
(b - c)
p<
Hunter 2
(a - c)
Stag (p)
Stag
Hare (1-p)
a, a
c,b
b,c
b, b
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Thus, if Hunter 1 thinks that Hunter 2 will hunt Stag with a
low probability
(b - c)
p<
Hunter 2
(a – c)
Then her safer choice is to
Stag (p)
Hare (1-p)
hunt Hare
Stag
a, a
c,b
b,c
b, b
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Thus, if players are unsure about the behavior of the other,
and they are rather pessimistic (they believe that the probability
that the other goes for Stag are low),
then the “individually safe”
Hunter 2
choice is to hunt Hare
Stag (p)
Stag
Hare (1-p)
a, a
c,b
b,c
b, b
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Example: Take a = 3, b = 2, and c = 1
Hunter 2
Stag (p)
Stag
Hare (1-p)
3, 3
1,2
2,1
2, 2
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Example: Take a = 3, b = 2, and c = 1
Then,
(b - c)
(2 – 1)
1
p<
=
=
(a– c)
(3 – 1)
2
Hunter 2
Stag (p)
Stag
Hare (1-p)
3, 3
1,2
2,1
2, 2
Hunter 1
Hare
Strategic Behavior in Business and Econ
Individually safe solution
Thus, if players believe that the other will hunt Stag with
a probability lower than 50%, the “individually safe” choice
is to hunt Hare
Hunter 2
Stag (p)
Stag
Hare (1-p)
3, 3
1,2
2,1
2, 2
Hunter 1
Hare
Strategic Behavior in Business and Econ
This game shows that some mutually beneficial outcome
may be hard to achieve if cooperation is required
The more pessimistic people is about the “social behavior” of
others, the harder is to achieve cooperation
Other examples of Stag hunt games are:
Two individuals who must row a boat
Raising funds for a public facility
The hunting practices of orca (known as carousel feeding)
are an example of a stag hunt. Here orcas cooperatively
corral large schools of fish to the surface and stun them
by hitting them with their tails. Since this requires that
fish not have any mechanism for escape, it requires the
cooperation of many orcas
Strategic Behavior in Business and Econ
The Volunteer's Dilemma
The Volunteer's Dilemma models the situation in which a public
(common) good must be provided.
While all the players benefit from the consumption of the public
good, each of them prefers the others to pay for it
Think of an scenario in which the electricity has gone out for a two
flat apartment building. The two inhabitants know that the
electricity company will fix the problem as long as at least one
person calls to notify them, at some cost. If no one volunteers, the
worst possible outcome is obtained for all participants. If any one
person elects to volunteer, the other benefits by not doing so
Strategic Behavior in Business and Econ
The environment of the game
Players:
Strategies:
Payoffs:
Person 1 and Person 2
Call or Don't call
(in the table)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Cooperation-Conflict
Symmetric
Strategic Behavior in Business and Econ
The Volunteer's Dilemma
Person 2
Call
Don't Call
Call
0, 0
0,1
Don't
Call
1,0
-10, -10
Person 1
Strategic Behavior in Business and Econ
The Volunteer's Dilemma
Person 2
We have two equilibria
Call
Don't Call
Call
0, 0
0,1
Don't
Call
1,0
-10, -10
Person 1
Strategic Behavior in Business and Econ
This is similar to a Game of Chicken in that the two
players prefer the other to Call
The difference is that Calling doesn't make you the
“Chicken”
Also, both Calling is not the best outcome for the
two of them
This example illustrates how difficult is the provision of
a public good.
Although everybody likes to have a City Park, all prefer
the other to pay for it
Strategic Behavior in Business and Econ
Hard to believe games
In some cases, the prediction of Game Theory seem to be highly
unintuitive. These are games that, if actually played, people doesn't
behave as expected
The Traveler's Dilemma
Guess 2/3 of the average game
Strategic Behavior in Business and Econ
The Traveler's Dilemma
An airline loses two suitcases belonging to two different travelers.
Both suitcases happen to be identical and contain identical
antiques. An airline manager tasked to settle the claims of both
travelers explains that the airline is liable for a maximum of $100
per suitcase, and in order to determine an honest appraised value of
the antiques the manager separates both travelers so they can't
confer, and asks them to write down the amount of their value at no
less than $2 and no larger than $100.
Strategic Behavior in Business and Econ
He also tells them that if both write down the same number, he will
treat that number as the true dollar value of both suitcases and reimburse both
travelers that amount. However, if one writes down a smaller number than the
other, this smaller number will be taken as the true dollar value, and both
travelers will receive that amount along with a bonus/malus: $2 extra will be
paid to the traveler who wrote down the lower value and a $2 deduction will
be taken from the person who wrote down the higher amount. The challenge
is: what strategy should both travelers follow to decide the value they should
write down
Strategic Behavior in Business and Econ
The environment of the game
Players:
Strategies:
Payoffs:
Traveler 1 and Traveler 2
Any (integer) number between 2 and 100
(in the table)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Conflict
Symmetric
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We can proceed by elimination of Dominated Strategies
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
This is the solution !
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
If we look for the Nash Equilibrium instead . . .
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
We draw the “Best Replies” (Red Circles)
Strategic Behavior in Business and Econ
The Traveler's Dilemma
Traveler 2
Traveler 1
The unique Nash Equilibrium is (2, 2) !
Strategic Behavior in Business and Econ
All “game theoretical” solutions have (2, 2) as the final
prediction of the game
Experimental evidence shows that players choose
quantities close to 100
Strategic Behavior in Business and Econ
Guess 2/3 of the average game
Guess 2/3 of the average is a game where several people guess
what 2/3 of the average of their guesses will be, and where the
numbers are restricted to the real numbers between 0 and 100. The
winner is the one closest to the 2/3 average.
0
100
Strategic Behavior in Business and Econ
Guess 2/3 of the average game
Example: Three players play the game and the choices
are: c1 = 42, c2 = 12, c3 = 23
0
12
23
42
c2
c3
c1
The average is ( 12 + 23 + 42 ) / 3 = 77 / 3 = 25.666
Thus, 2/3 of the average is 2/3 of 77/3 = 154/9 = 17.11111
Strategic Behavior in Business and Econ
100
17.1111 25.666
0
12
23
42
c2
c3
c1
c1 – 17.111 = 42 – 17.111 = 24.888
c2 – 17.111 = 12 – 17.111 = -5.111111
c3 – 17.111 = 23 – 17.111 = 5.888
In this example, player 2 is the winner with c2 = 12
Strategic Behavior in Business and Econ
100
The environment of the game
Players:
Strategies:
Payoffs:
Player 1, Player 2, Player 3, ...
Any number between 0 and 100
$1 if win, 0 otherwise. If there is a tie,
the dollar is split among the winners
(with respect to previous examples, this game features more
than 2 players and an infinite number of strategies !)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Conflict
Symmetric
Strategic Behavior in Business and Econ
Guess 2/3 of the average game (Analysis)
Although this game doesn't admit a “table” representation,
it can be solved by elimination of Dominated Strategies
Notice first that for whatever choices of a number by the
players, the average can NEVER be larger than 100
Therefore, 2/3 of the average can NEVER be above
(2/3) x 100 = 66.67
Therefore, I should NEVER choose a number above 66.67
That is, the choice of any number above 66.67 is a
Dominated Strategy
Strategic Behavior in Business and Econ
Guess 2/3 of the average game (Analysis)
Although this game doesn't admit a “table” representation,
it can be solved by elimination of Dominated Strategies
Notice first that for whatever choices of a number by the
players, the average can NEVER be larger than 100
Therefore, 2/3 of the average can NEVER be above
(2/3) x 100 = 66.67
Therefore, I should NEVER choose a number above 66.67
That is, the choice of any number above 66.67 is a
Dominated Strategy
Dominated Strategies
0
66.67
Strategic Behavior in Business and Econ
100
Guess 2/3 of the average game (Analysis)
Hence, the players will never choose a number above 66.67
and then the average can NEVER be larger than 66.67
Therefore, 2/3 of the average can NEVER be above
(2/3) x 66.67 = 44.45
Therefore, I should never choose a number above 44.45
That is, the choice of any number above 44.45 is now a
Dominated Strategy
Dominated Strategies
0
66.67
Strategic Behavior in Business and Econ
100
Guess 2/3 of the average game (Analysis)
Hence, the players will never choose a number above 66.67
and then the average can NEVER be larger than 66.67
Therefore, 2/3 of the average can NEVER be above
(2/3) x 66.67 = 44.45
Therefore, I should never choose a number above 44.45
That is, the choice of any number above 44.45 is now a
Dominated Strategy
Dominated Strategies
0
44.45
66.67
Strategic Behavior in Business and Econ
100
Guess 2/3 of the average game (Analysis)
Hence, the players will never choose a number above 44.45
and then the average can NEVER be larger than 44.45
Therefore, 2/3 of the average can NEVER be above
(2/3) x 44.45 = 29.64
Therefore, I should never choose a number above 29.64
That is, the choice of any number above 29.64 is now a
Dominated Strategy
Dominated Strategies
0
44.45
66.67
Strategic Behavior in Business and Econ
100
Guess 2/3 of the average game (Analysis)
Hence, the players will never choose a number above 44.45
and then the average can NEVER be larger than 44.45
Therefore, 2/3 of the average can NEVER be above
(2/3) x 44.45 = 29.64
Therefore, I should never choose a number above 29.64
That is, the choice of any number above 29.64 is now a
Dominated Strategy
Dominated Strategies
0
29.64
44.45
66.67
Strategic Behavior in Business and Econ
100
Guess 2/3 of the average game (Analysis)
And by continuing the elimination of Dominated
Strategies, the only choice that survives is to
choose 0 as your guess of 2/3 of the average
of all the number chosen by the players
Dominated Strategies
0
29.64
44.45
66.67
Strategic Behavior in Business and Econ
100
Guess 2/3 of the average game (Analysis)
The predicted outcome is that all the players will choose 0.
Then, the average will be 0, and 2/3 of the average will
also be 0. That is, all players win.
Experimental evidence goes against this theoretical prediction
In an Internet contest with 19.196 people playing and a $865 prize,
the winning value was 21.6
Strategic Behavior in Business and Econ
Hard to analyze games
Most games that represent real business and/or economics
scenarios often involve a large number of strategies, many
players, and complicated rules. In such cases a simple table
representation is not possible and the search for equilibria
requires other techniques
Hotelling Spatial Competition Game
Strategic Behavior in Business and Econ
Hotelling Spatial Competition Game
Suppose that there are two competing shops selling the same
indistinguishable product at the same price, and that they must be
located along the length of a street running north and south. Each
shop owner wants to locate his shop such that he maximizes his
own market share by drawing the largest number of customers.
Customers are spread equally along the street. Suppose, finally, that
each customer will always choose the nearest shop
1
2
Strategic Behavior in Business and Econ
The environment of the game
Players:
Strategies:
Payoffs:
Shop 1 and Shop 2
Any location within the length of the street
For each shop, all the customers that are closer
to it than to the competitor
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Strategic Behavior in Business and Econ
Simultaneous
Conflict
Symmetric
What will be the location of the shops ?
1
2
Strategic Behavior in Business and Econ
First Guess . . .
1
2
Strategic Behavior in Business and Econ
But, what would you do if you were Shop 1
1
2
Strategic Behavior in Business and Econ
But, what would you do if you were Shop 1
1
Strategic Behavior in Business and Econ
2
You get more costumers than before !
1
Strategic Behavior in Business and Econ
2
What will be the “best reply” by Shop 2 ?
1
Strategic Behavior in Business and Econ
2
What will be the “best reply” by Shop 2 ?
2
1
Strategic Behavior in Business and Econ
You get more costumers than before !
2
1
Strategic Behavior in Business and Econ
If this dynamics continues, the unique stable point is
1
2
Strategic Behavior in Business and Econ
The two shops locate at the middle of the street !
1
2
Strategic Behavior in Business and Econ
This is an example of “spatial clustering” of business
There are real examples of this
Car Dealers
Oriental rug stores
Computer (electronics) districts
Michigan Avenue
But this also may apply to other characteristics of
businesses:
Quality
Sweetness
Even to politics !
Strategic Behavior in Business and Econ
This model can also explain the common complaint that, for instance, the
presidential candidates of the two American political parties are
"practically the same". Once each candidate is confirmed during
primaries, they are usually established within their own partisan camps.
The remaining undecided electorate resides in the middle of the political
spectrum, and there is a tendency for the candidates to "rush for the
middle" in order to appeal to this crucial bloc. Like the paradigmatic
example, the assumption is that people will choose the least distant
option, (in this case, the distance is ideological) and that the most votes
can be had by being directly in the center.
Strategic Behavior in Business and Econ
Primaries
Strategic Behavior in Business and Econ
Nominees
Strategic Behavior in Business and Econ
Final campaign
Strategic Behavior in Business and Econ
What would happen if . . .
Strategic Behavior in Business and Econ
What would happen if . . .
This is know as the Median Voter Theory, very popular in
modern Political Science
Strategic Behavior in Business and Econ