Transcript Rational Choice Sociology

Rational Choice
Sociology
Lecture 6
Game Theory II: Some 2-person
Non-Cooperative Non-Zero Sum
Games
Concept of Game
Solution
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Game theory not only describes and classifies strategic
situations, but also elaborates methods how to solve the games
Game solution is answer to the question: given general
assumptions of game theory and description of situation
(number of players, their choice possiblilities, outcomes and
their utilities for different players, which strategies actors will
choose, which outcome will take the place? What will happen?
The outcome predicted by game theory is solution.
Each game solution is game equilibrium (stable outcome), but
not each equilibrium is game solution. If game has only one
equilibrium outcome, this outcome is the solution of game.
However, some games have more than one equilibrium outcome.
The outcome is equilibrium (or stable outcome), if no player in
this outcome can improve her welfare by one-sided actions (she
can make herself only worse if she would choose differently
while other players do not change their choices)
There is different kind equilibria depending on the type of the
game and the kind of reasoning used to identify these equilibria
Prisoner’s Dilemma I
Actor 2
Actor 1
C
D
C
3 CC 3
1 CD
D
4 DC 1
4
2 DD 2
Actor 1: DC > CC > DD > CD
4
3
2
1
Actor 2: CD > CC > DD > DC
4
3
2
1
Prisoner’s Dilemma II
Why Prisoner’s Dilemma is “nice”
game (although about nasty
situations)?
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It has simple and strong solution – can be solved in
dominant strategies: there is outcome that is dominant
strategy equilibrium (DSE)
It discloses the logic of situation characteristic for the
wide range of situations where people can gain by
cooperation but face the obstacles to realization of these
gains that derive from their own rationality
Solution in dominant
strategies
A game can be solved in dominant strategies, if at
least one player has dominating strategy (as
explained in the lecture 3: the choice that is better
no matter other actor chooses).
In the Prisoner’s Dilemma game, both players have
dominating strategies (defection): so the game has
solution (outcome DD) that is dominant strategy
equilibrium (DSE)
However, to solve a game in dominant strategies, it is
sufficient that only one player has dominant
strategy
Prisoner’s Dilemma in the
Real Social Life
Most important cases of prisoner’s dilemma in real social life are
include collective action to produce some public good
Public good is a good that can be used also by the actors that have
not participated in the cost of creating it
In situations where the welfare can be enhanced by production of
public good have the incentive to take free ride (zuikiauti,
išsisukinėti) and exploit those who pay the production cost
(=suckers)
To build completely precise model of such situation, one should use
N-person prisoner’s dilemma game. However, as approximate
(simplifying) model a 2-person game is sufficient, where one
person plays against an Other(s) (either from remaining)
Prisoner’s Dilemma in the Real Social Life :
examples I
Public procurement contest is announced – whether John
should/would bribe officials to win? (X means dominant strategy)
Others
John
Do not bribe
(c)
Do not bribe (C)
XBribe (D)
3 CC 3
4 + DC 1
Actor 1: DC >
4
Actor 2: CD >
4
XBribe (D)
1 CD
4-
2+ DD 2-
CC > DD > CD
3
2
1
CC > DD > DC
3
2
1
Prisoner’s Dilemma in the Real Social Life : examples
II
There is no control in the public transport: would
John buy a thicket
Others
John
Buy (C)
Buy (C)
X Do not buy
3 CC 3
4+ DC 1
X Do not buy
1 CD
4-
2 + DD 2-
Actor 1: DC > CC > DD > CD
4
3
2
1
Actor 2: CD > CC > DD > DC
4
3
2
1
Prisoner’s Dilemma in the Real Social Life : examples
III
There is no tax payment control: would John pay
taxes?
Other
John
Pay taxes(c)
(X) Do not pay
taxes(D)
Pay taxes (c)
(X) Do not pay
taxes(D)
3 CC 3
1 CD
4 + DC 1
4-
2+ DD 2-
Actor 1: DC > CC > DD > CD
4
3
2
1
Actor 2: CD > CC > DD > DC
4
3
2
1
Prisoner’s Dilemma in the Real Social Life : examples IV
What John should do with used handkerchief?
Other
To put into
John
litterbin
To put into
litterbin (c)
(X) Throw away
(D)
(c)
3 CC 3
4+ DC 1
(X) Throw
away (D)
1 CD
4-
2- DD 2-
Actor 1: DC > CC > DD > CD
4
3
2
1
Actor 2: CD > CC > DD > DC
4
3
2
1
Prisoner’s Dilemma in the Real Social Life : examples
V
The neighbourhood is vandalize by the gang of young delinquents. Who will
complain to police?
Other
John
Inform(c)
X Do not inform
(D)
Inform(c)
X Do not
inform (D)
3 CC 3
1 CD
4+ DC 1
4-
2- DD 2-
Actor 1: DC > CC > DD > CD
4
3
2
1
Actor 2: CD > CC > DD > DC
4
3
2
1
Chicken Game (or: Hawk Dove Game) I
Two drivers drive towards each other on a collision course: one must swerve, or both may
die in the crash, but if one driver swerves and the other does not, the one who swerved
will be called a "chicken," meaning a coward. While each player prefers not to yield to the
other, the outcome where neither player yields is the worst possible one for both players.
Many wars started in the Chicken Game situation.
Actor 2
C
Actor 1
C (turn)
D (do not
turn)
(turn)
3 CC 3
4+ DC 2-
Actor 1: DC > CC > CD >
4
3
2
Actor 2: CD > CC > DC >
4
3
2
D
(do not
turn)
2+ CD
4-
1 DD 1
DD
1
DD
1
The Concept of Nash equilibria
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In the Chicken game situations, no actor has dominant strategy.
Therefore, this game cannot be solved in dominant strategies (has no
DSE)
However, it has two Nash equilibria (so called to honor American
mathematician John Nash (Nobel prize in economics 1994, subject of
Hollywood movie “Beautiful mind”).
Nash equilibrium is the game outcome that is the result of choices that
are best replies to each other.
Searching for Nash equilibrium, we consequently take the position of
each actor, assume the choices of her partner as given and ask, which
reply would be the best for her? Then take the position of other actor and
do the same for her.
If there is dominating strategy, all the best answer will be in the same
raw or column
Outcomes, which result from best replies, are Nash equilibria
If there is only one Nash equilibrium outcome, this outcome is the
solution of game
If there are more than one Nash equilibriium, after finding them we have
important information about the game, but the game is still not solved
Solution of Chicken Game in Mixed Strategies:
Actor 1 C 99/100; D 1/100
Actor 2 C 99/100; D 1/100
Actor 2
C
Actor 1
C (turn)
D (do not
turn)
(turn)
0 CC 0
+1+ DC -1-
Actor 1: DC > CC > CD >
4
3
2
Actor 2: CD > CC > DC >
4
3
2
D
(do not
turn)
-1+ CD
+1-
-100 DD -100
DD
1
DD
1
The search for Nash equilibria (an
example) (+ to the right from number means best move for
row player; - to the right of number means best move for column
player)
S1
S2
S1
4
2
12
S2
-2
6
33 +
S3
3
0
-8
S4
9+
S5
4
1
5
11-
S6
2
5
0
0
2
31
8
1
4
2
S3
S4
-2
99-
16
4
5
23
S5
1
7
8
8-
-3
-5
6
7
14
6
69+
8
9-
3
9-
21
1
22 +
-5
12
4
-1
0
8
7-
56 +
0
12
96-
4
The concept of Pareto
optimum
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To solve some games with multiple Nash equilibria, the concept
of Pareto optimal outcome is helpful
The concept of Pareto optimum (so called to honour the great
Italian economist and sociologist Vilfredo Pareto 1848-1923) is
the definition of collective welfare preferred by economists.
The state X is Pareto optimal if there are no Pareto-superior
states with respect to state X
State Y is Pareto-superior with respect to state X if at least on
actor favors state Y, and the remaining actors are indifferent (in
other words: Y is Pareto superior with respect to X if at least
one actor gains after transition from X to Y or votes for such
transition)
Pareto improvements involve no redistribution; they are
improvements at nobody’s expense; they are voted for
unanimously or with nobody voting against
The states X and Y are Pareto-equivalent if nobody nothing gains
and nothing loses if X is exchanged for Y or vice versa (all actors
are indifferent; all abstain from voting)
The states X and Y are Pareto incomparable if at least one actor
is worse if X is exchanged for Y or vice versa (at least one actor
votes against)
Pareto optimality and
stability of outcomes
• There is no definite relation between the property
of an outcome to be Pareto-optimal and to be an
equilibrium
An outcome can be an equilibrium, but Pareto suboptimal or be Pareto-optimal, but not an equilibrium
(this is the case in the Prisoner’s Dilemma game)
A game can have a lot Pareto optimal outcomes, but
none of then can be equilibrium states
Usually there are many Pareto-optimal states that are
Pareto-incomparable
How the concept of Pareto-optimality can
help solve a game:
If there are 2 or more Nash equilibria in the game, and one of them
is Pareto superior, than the Pareto optimal equilibrium is the
solution of game (in the example –Assurance game )
Actor 2
Actor 1
C
D
C
D
4 + CC 4-
1 CD
3 DC 1
3
2 + DD 2-
Actor 1: CC> DC > DD > CD
4
3
2
1
Actor 2: CC > CD > DD > DC
4
3
2
1
Stag hunt (very similar to Assurance game):
Two hunters can either jointly hunt a stag (an adult deer and rather large
meal) or individually hunt a rabbit (tasty, but substantially less filling).
Hunting stags is quite challenging and requires mutual cooperation. If either
hunts a stag alone, the chance of success is minimal.
Actor 2
Actor 1
Hunt stag
Hunt hare
Hunt stag
Hunt hare
20 + SS 20-
-3 SH
5 HS
-3
Actor 1: CC> DC = DD > CD
3
2
2
1
Actor 2: CC > CD = DD > DC
3
2
2
1
5
5+ HH 5-
Coordination games
Assurance game is subtype of the non-cooperative non-zero
sum games that are called in game theory coordination
games. In such games, actors gain most if, if they choose
identical actions. This differentiates them from games
like Prisoner’s Dilemma and Chicken game, when the
greatest payoffs are associated with the outcomes
resulting from different actions choice (the actors wins
most when she free rides while other agent cooperates).
In coordination games, there is no possibility to win most
choosing differently from other actor.
The problem is that there are many different ways to
coordinate actions (many Nash equilibria). Paretooptimality helps solve some coordination games: where
one from two or more equilibria is Pareto-superior.
However, Pareto optimality cannot help if all equilibria are
Pareto equivalent or Pareto incomparable
Heads or Tails (another name: Same or Different) game:
Coordination game with Pareto equivalent equilibria
Jonas and Petras are in different rooms. They cannot communicate. To
broker, they should say “Head” or “Tail”. If they say the same word, they win
1 litas each. If they say different words, they win nothing
Petras
Jonas
Head
Tail
Head
Tail
1 + HH 1-
0H0
0 TH 0
Jonas: HH = TT > TH = HT
2
2
1
1
Petras: HH = TT > TH = HT
2
2
1
1
1+ TT 1-
The Concept of Focal Point
Labor experiments show that people say more frequently “Head”
playing “Same or Different (Heads or Tails)” game. Why?
Thomas Schelling (born 1921, Nobel Memorial prize in economics) explains this
phenomenon using the concept of focal point: focal point is equilibrium in
coordination game that has some distinctive features in the context of the
culturally shared (common) knowledge by all players
Real life situations where people play “Same or Different” game with many equilibria:
You lost your friend after arriving to foreign city. There are no arrangements where you will meet in such
situation, no hotel booked and no possibility of communication. Where you will wait for your friend in
1)
Paris
2)
Vilnius
3)
Moscow
4)
Berlin
What would be findings of a survey?
The place named by most respondents is focal point (each place in Vilnius ir Paris is equilibrium, but only
few focal points)
However, some analysts find the concept unsatisfactory because of its unformal, half-intuitive or empirical
character (solution is find by survey, not by deductive argument!).
Coordination games with Pareto-equivalent equilibria are trivially solved if description of game is changed,
allowing to communicate. This is not necessary the case in coordination games where are Paretoincomparable Nash equilibria
Battle of Sexes: Coordination game with Pareto incomparable equilibria:
Imagine a couple that would like to spend the evening, but cannot agree whether to
attend the opera or a football match. The husband would most of all like to go to the
football game. The wife would like to go to the opera. Both would prefer to go to the same
place rather than different ones (therefore, the game is coordination game); 2 versions –
with communication or without.
Mary
John
Football
Opera
John:
Mary:
Football
Opera
4 + FF 3-
2 FO 2
1 OF 1
FF > OO > FO > OF
4
3
2
1
OO > FF > FO > OF
4
3
2
1
3+ OO 4-
Anti-Coordination game: constant-sum game with no equilibria
without Nash equilibria
Jonas and Petras are in different rooms. They cannot communicate. To
broker, they should say “Head” or “Tail”. If they say the same word, then
Jonas wins 1 litas. If they say different words, then Petras wins 1 litas. (or:
Jonas should guess, what Petras says; if his guess is correct, he wins;
otherwise, Petras wins)
Petras
Jonas
Head
Tail
Head
Tail
1 + HH 0
0 HT 1-
0 TH 1-
Jonas:
Petras:
HH = TT > TH = HT
2
2
1
1
TH = HT > HH =TT
2
2
1
1
1 +TT 0
The Concept of Mixed Strategy
If game has no Nash equilibrium or multiple Pareto-equivalent or Paretoincomparable equilibria, it cannot be solved in pure strategies, but may
be solvable in mixed strategies (have mixed strategies equilibrium)
Strategy is pure if it is not mixed.
Strategy is mixed, if it is chosen probabilistically. In this case, there are 2
choice stages:
(1)
Choice between probability distributions
(2)
Choice between pure strategies using chosen probability distributions.
E.g. in the anticoordination game, Jonas and Petras can choose between
infinitely many probability distributions: 1) With probability 1/3 say
“Heads”, with probability 2/3 say “Tails”; 2) With probability 2/5 say
“Heads”, with probability 3/5 say “Tails” etc.
After selecting one distribution, player uses the device generating events
with chosen probability distribution to choose “Heads” or “Tails”.
In the Anti-Coordination game mixed strategy equilibrium is the outcome of
the strategy choice where Jonas and Petras each choose “Heads” with
probability 0,5 and “Tails” with probability 0,5
The finding of the mixed strategy equilibrium may be mathematically
challenging, if there are many strategies to choose. Besides, the
magnitudes of payoffs maybe important (if utility indexes are cardinal)
Chicken Game solution in mixed strategies
Actor 1: C p=99/100 D p=1/00
Actor 2 C p=99/100 D p=1/100
Actor 2
C
Actor 1
C (turn)
D (do not
turn)
(turn)
0 CC 0
+1+ DC -1-
Actor 1: DC > CC > CD >
4
3
2
Actor 2: CD > CC > DC >
4
3
2
D
(do not
turn)
-1+ CD
+1-
-100 DD -100
DD
1
DD
1
Iterated games
• Mixed strategy equilibria solutions make sense for
iterated games
• The equilibria in iterated games can be different
from equilibria in one-shot games
E.g. In one-shot Prisoner’s Dilemma the game solution
is DD.
In iterated PD (under assumption that there is no
knowledge which episode of the game will be the
last one), the solution of the game is Tit or Tat:
Cooperate, the continue to Cooperate, if other
player Cooperates; if other actor Defects, Defect
(the actors cooperate because of the shadow of
future).
Sequential games
• Some games can be easily solved, if the assumption is
added, that players make their moves not simultaneously,
but in turn and know, what was the choice of their
partner
However, to analyze such games the matrix or table models
of the games are not sufficient. One should use decision
tree-types diagrams.
Sequential Battle of Sexes game; John
chooses first
Football
Mary
FF; 4, 3 X
Football
Opera
Mary
John
OF; 2,2
Football
OF; 1, 1
Opera
Opera
OO; 3,4
Sequential Battle of Sexes game; Mary
chooses first
Football
John
FF; 4, 3
Football
Opera
John
OF; 1.1
Football
OF; 2, 2
Mary
Opera
Opera
OO; 3,4 X