Rational Choice Sociology

Lecture 6Game Theory II: Some 2-

person Non-Cooperative Non-Zero Sum Games

Concept of Game Solution

• 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

Actor 1: DC > CC > DD > CD 4 3 2 1

Actor 2: CD > CC > DD > DC 4 3 2 1

C D

C 3 CC 3 1 CD 4

D 4 DC 1 2 DD 2

Prisoner’s Dilemma II

Why Prisoner’s Dilemma is “nice” game (although about nasty situations)?

• 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 strategiesA 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 LifeMost 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 IPublic procurement contest is announced – whether John should/would bribe officials to win? (X means dominant strategy)

Others

John

Actor 1: DC > CC > DD > CD 4 3 2 1

Actor 2: CD > CC > DD > DC 4 3 2 1

Do not bribe (c) XBribe (D)

Do not bribe (C) 3 CC 3 1 CD 4-

XBribe (D) 4 + DC 1 2+ DD 2-

Prisoner’s Dilemma in the Real Social Life : examples IIThere is no control in the public transport: would John buy a thicket

Others

John

Actor 1: DC > CC > DD > CD 4 3 2 1

Actor 2: CD > CC > DD > DC 4 3 2 1

Buy (C)X Do not buy

Buy (C) 3 CC 3 1 CD 4-

X Do not buy 4+ DC 1 2 + DD 2-

Prisoner’s Dilemma in the Real Social Life : examples IIIThere is no tax payment control: would John pay taxes?

Other

John

Actor 1: DC > CC > DD > CD 4 3 2 1

Actor 2: CD > CC > DD > DC 4 3 2 1

Pay taxes (c)(X) Do not pay taxes(D)

Pay taxes(c) 3 CC 3 1 CD 4-

(X) Do not pay taxes(D)

4 + DC 1 2+ DD 2-

Prisoner’s Dilemma in the Real Social Life : examples IVWhat John should do with used handkerchief?

Other

John

Actor 1: DC > CC > DD > CD 4 3 2 1

Actor 2: CD > CC > DD > DC 4 3 2 1

To put into

litterbin (c)

(X) Throw away (D)

To put into litterbin (c)

3 CC 3 1 CD 4-

(X) Throw away (D)

4+ DC 1 2- DD 2-

Prisoner’s Dilemma in the Real Social Life : examples VThe neighbourhood is vandalize by the gang of young delinquents. Who will complain to police?

Other

John

Actor 1: DC > CC > DD > CD 4 3 2 1

Actor 2: CD > CC > DD > DC 4 3 2 1

Inform(c)X Do not inform (D)

Inform(c) 3 CC 3 1 CD 4-

X Do not inform (D)

4+ DC 1 2- DD 2-

Chicken Game (or: Hawk Dove Game) ITwo 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

Actor 1

Actor 1: DC > CC > CD > DD 4 3 2 1

Actor 2: CD > CC > DC > DD 4 3 2 1

C (turn) D (do not turn)

C (turn) 3 CC 3 2+ CD 4-

D (do not turn)

4+ DC 2- 1 DD 1

The Concept of Nash equilibria

• 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

Actor 1

Actor 1: DC > CC > CD > DD 4 3 2 1

Actor 2: CD > CC > DC > DD 4 3 2 1

C (turn) D (do not turn)

C (turn) 0 CC 0 -1+ CD +1-

D (do not turn)

+1+ DC -1- -100 DD -100

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 S3 S4 S5

S1 4 2 12 8 -2 99- 16 1 7 8

S2 -2 6 33 + 1 4 5 23 8- -3 -5

S3 3 0 -8 4 6 7 14 6 69+ 96-

S4 9+ 2 31 2 8 9- 3 9- 21 1

S5 4 1 5 11- 22 + -5 12 4 -1 0

S6 2 5 0 0 8 7- 56 + 0 12 4

The concept of Pareto optimum• 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 equilibriumAn outcome can be an equilibrium, but Pareto sub-optimal

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

Actor 1: CC> DC > DD > CD 4 3 2 1

Actor 2: CC > CD > DD > DC 4 3 2 1

C D

C 4 + CC 4- 1 CD 3

D 3 DC 1 2 + DD 2-

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

Actor 1: CC> DC = DD > CD 3 2 2 1

Actor 2: CC > CD = DD > DC 3 2 2 1

Hunt stag Hunt hare

Hunt stag 20 + SS 20- -3 SH 5

Hunt hare 5 HS -3 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). Pareto-optimality 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 equilibriaJonas 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

Jonas: HH = TT > TH = HT 2 2 1 1

Petras: HH = TT > TH = HT 2 2 1 1

Head Tail

Head 1 + HH 1- 0 H 0

Tail 0 TH 0 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 in1) Paris2) Vilnius 3) Moscow4) BerlinWhat 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 Pareto- incomparable 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

John: FF > OO > FO > OF 4 3 2 1

Mary: OO > FF > FO > OF 4 3 2 1

Football Opera

Football 4 + FF 3- 2 FO 2

Opera 1 OF 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

Jonas: HH = TT > TH = HT 2 2 1 1

Petras: TH = HT > HH =TT 2 2 1 1

Head Tail

Head 1 + HH 0 0 HT 1-

Tail 0 TH 1- 1 +TT 0

The Concept of Mixed Strategy

If game has no Nash equilibrium or multiple Pareto-equivalent or Pareto-incomparable 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 strategiesActor 1: C p=99/100 D p=1/00Actor 2 C p=99/100 D p=1/100

Actor 2

Actor 1

Actor 1: DC > CC > CD > DD 4 3 2 1

Actor 2: CD > CC > DC > DD 4 3 2 1

C (turn) D (do not turn)

C (turn) 0 CC 0 -1+ CD +1-

D (do not turn)

+1+ DC -1- -100 DD -100

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

Mary

JohnMary

Football

Opera

Football

Opera

Football

Opera

FF; 4, 3 X

OF; 2,2

OF; 1, 1

OO; 3,4

Sequential Battle of Sexes game; Mary chooses first

John

MaryJohn

Football

Opera

Football

Opera

Football

Opera

FF; 4, 3

OF; 1.1

OF; 2, 2

OO; 3,4 X