game theory game theory analyses how rational decisions are made when decisions are interactive:...
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Game Theory
Game Theory analyses how rational decisions are made when decisions are interactive: This means that each individual’s best choice depends on the choices made by other individuals.
In the conventional model of the market economy, each individual’s best choice depends only on that individual’s preferences and costs, it does not depend on the choices made by others.
Thus in the cases discussed by game theory, each individual’s perception or belief about what others might do has a direct impact on that person’s choice. Beliefs and perceptions now matter, in a way they do not in conventional economic theory.
The Elements of Game Theory 1. The players. These are the individuals, i, who make decisions. Each player's goal is to maximize his/her utility by a choice of actions.
2. The actions of each player ai are the choices they can make.
3. The information set for each player is the knowledge he/she has at each stage of the game of the values of different variables.
4. A player's strategy si are the actions player i chooses at each stage of
the game, given his/her information set.
5. A player's payoff Πi(s1,s2,..sn) is the expected utility of the player given
the strategies chosen by him/herself and all other players.
6. The equilibrium s*=(s1*,...,sn*) is a strategy combination consisting of
a best strategy for each of the n players in a game. There are a number of different equilibrium concepts in game theory which we will discuss in turn.
Game theory versus Conventional Economics
• Game theory tells us that rational individuals know that other individuals are also rational and will do what is in their best interest so each individual will choose an action that maximizes their benefit given that other individuals will be doing the same
• Paradoxically this gives very different results from market economic theory where individual maximization always results in social maximization
• In game theory social maximization does not always follow from individual maximization
The core components of Game theory
• A game is a set of rules known to all players according to which the players can get different payoffs by selecting different strategies.
• To make the modelling tractable, the rules themselves are fixed and cannot be changed by the players
• However, the payoff for each player depends on the choices of other players
• This makes game theory different from the theory of individual choice in conventional economics where the choice made by an individual does not depend on choices made by other individuals (except in so far as the latter determine the prices faced by the individual when making his/her choice)
• Game theory analyses how rational individuals will make their choices in such an interactive context
Cooperative and Non-Cooperative Games
Cooperative games are those where players can make binding agreements with each other about the strategies they will play.
In contrast, non-cooperative game theory assumes that individuals will not operate in any way which is not individually rational.
Thus “cooperative games” assume effective external enforcement of contracts. “Non-cooperative games” assume contracts cannot be enforced unless the individual wants the contract to be enforced.
Game theory is for the most part about non-cooperative games. Cooperative games should not be confused with “cooperation” in non-cooperative games where individuals can cooperate with each other because that is the best strategy for them in that game.
Equilibrium Concepts in Game Theory
A number of equilibrium concepts are used in game theory. The best known are the Dominant Strategy and Nash Equilibrium concepts.
In discussing equilibria we use the notation s-i to denote the vector of
strategies of all players other than i.
Player i's best response to the strategies s-i chosen by other
players is the strategy s* that yields the greatest payoff,
π i(si*,s-i) ≥ π i(si', s-i) ▼si' ≠ si*
The strategy si* is a dominant strategy if it is a player's strictly best
response to any strategies the other players may pick, in the sense that whatever strategies they pick, his payoff is highest with si*,
πi(si*,s-i) ≥ πi(si',s-i) for ▼s-i, and ▼si' ≠ si*
The Nash Equilibrium Concept
The strategy combination s* is a Nash Equilibrium if no player has an incentive to deviate from his strategy given that the other players do not deviate,
▼i, πi(si*,s-i*) ≥ πi(si',s-i*) ▼si'.
Note that the definition of Nash equilibrium lacks the extra ▼s-i of the
dominant strategy equilibrium so a Nash strategy only needs to be the best response to other Nash strategies, not to all possible strategies. This is also shown in the definition by the fact that the Nash maximizes πi(si*,s-i*) and not πi(si*,s-i).
Thus the Nash is a best response on the assumption that other players are playing Nash. Hence Varian argues that the Nash equilibrium is an equilibrium in actions and beliefs.
The Nash equilibrium
• In the Nash equilibrium each person asks:• If I follow a particular strategy, assuming that others are
following a particular strategy, will they continue to follow their particular strategies if they know that I am following this strategy?
• If the answer to this question is yes for every individual following a strategy, the combination of strategies for all individuals is a Nash equilibrium for the game
• The characteristic of a Nash equilibrium is that once everyone is playing a Nash, no-one will play any other strategy, even if the Nash does not appear rational for all players collectively
• There may be more than 1 Nash equilibrium for a game, and the outcome may be socially undesirable because with more than one equilibrium, rational players can make mistakes and end up with suboptimal outcomes
Justifications for the Nash Equilibrium Concept
1. It gives a formal definition of equilibrium if there is a unique self-evident way to play a game. If there is an equilibrium to a game, it must be because rational players have no incentive to move away from it.
2. The Nash may represent a self-enforcing convention. If the players can engage in (non-binding) communication before the game then they will have to work out a strategy combination which is Nash.
However, the Nash has a number of problems:1. There may be more than one possible Nash equilibrium. Game theory does not tell us how equilibria are selected. 2. The Nash equilibrium solution may be so complex that it is difficult to argue that the solution is the self-evident equilibrium which would be chosen by rational actors.
3. The Nash simply describes the likely conventions which emerge. It does not explain the emergence of beliefs about what other agents will do. It is thus descriptive rather than analytical.
The Dominant Strategy equilibrium
• Some games (like the prisoner’s dilemma) have a stronger equilibrium called the Dominant Strategy equilibrium
• Here, each player asks if there is a strategy for that player that is the best strategy regardless of the particular strategies played by the other players.
• If it exists, such a strategy is called a dominant strategy for that player
• If all players have a dominant strategy, the game has a dominant strategy equilibrium, and these equilibria are very stable
• All dominant strategy equilibria are also Nash equilibria, but not vice versa
The Payoff Structure in a Simple Game
• In a 2 person game we write the Row player’s payoff first, the Column player’s payoff second
A
B
Drive on Left Drive on Right
Drive on Left
Drive on Right
1,1
1,1
-10,-10
-10,-10
How will rational individuals decide what to do?
Nash Equilibria in a Coordination Game
A
B
Drive on Left Drive on Right
Drive on Left
Drive on Right-10,
1, 1 ,-10Nash Equilibrium
-10Not NE
-10Not NE
1
Nash Equilibrium
Thus in this simple coordination game there are 2 Nash equilibria and rational individuals can make mistakes ending up with -10,-10
There are no dominant strategies in this game
1,
Extensive Form of (Coordination Game)
B
A
B
1,1-10,-101,1 -10,-10
Identifying Nash Equilibrium
B
Left
Right
Left Right
A
0,0
5,1
9,-1
4,4
Take each strategy combination like (Left, Right), and ask i) will A play Left if B plays Right and ii) will B play Right if A plays Left. If the answers to both questions are YES, (Left, Right) is a Nash equilibrium
Do this for all possible strategy combinations
The Prisoner’s Dilemma Game
The most widely studied problem involved in game theory is that of free-riding. This is the well known Prisoner's Dilemma model.
Free-riding is a structure of payoffs where there are gains from cooperation but where the individual gains are even greater if everyone cooperates but the individual cheats.
With Prisoner's Dilemma payoffs the private incentive structure creates no incentive for cooperative behaviour and mutually gainful cooperation now requires stronger conditions. The Nash equilibrium here is not the pareto efficient one.
The Prisoner's Dilemma problem emerges when the payoff structure is such that i) cooperation is the socially optimal strategy, so unlike the Chicken Game, there is a unique outcome which is socially desirable, ii) the best outcome for each individual is that the other person cooperates and he defects, iii) the worst outcome is that he cooperates and the other defects.
The Payoff Structure in a Prisoner’s dilemma
A
B
Cooperate Defect
Cooperate
Defect
8,8
1,1
0,10
10,0
What matters is not the actual payoffs but their ranking for each player
Highest joint payoff
Highest payoff for A but lowest payoff for B
Lowest payoff for A but highest payoff for B
Lowest joint payoff
Nash Equilibrium in the Prisoner’s dilemma
A
B
Cooperate Defect
Cooperate
Defect10,
8, 8 ,10Not Nash Equilibrium
0Not NE
0Not NE
1, Nash Equilibrium
1
Defect, Defect is not only a Nash equilibrium, it is a Dominant Strategy Equilibrium as well and is therefore very stable
Prisoner’s Dilemma 2
Non-cooperation now becomes the Dominant Strategy, ie whatever anyone else does, the best strategy for each individual is to defect, or choose the socially sub-optimal option.
Examples of prisoner’s dilemma games:
i) The Creation of Social Order: Hobbes and the Theory of the Leviathan
ii) Cartels: Adam Smith and the Free-Market Doctrine
iii) The Provision of Public Goods
iv) Worker-Capitalist Cooperation: Leibenstein and the Explanation of Differences in Capitalist Performance based on Trust
v) As a component of all types of collective action problem
Escapes from the dominant strategy equilibrium
• Rational individuals will harm themselves by being forced into the dominant strategy equilibrium by their fear of being ripped off by other rational individuals
• There are three escapes from the prisoner’s dilemma• i) External enforcement. This is where the role of the state
comes in as external enforcer. Weber defined the state as the body with a legitimate monopoly of violence
• ii) Repeated games. If the same players play the game for an indefinite number of times, cooperation may emerge as a voluntary strategy if some players can credibly threaten to punish those who defect by not playing cooperate strategies with them
• iii) Less than rational players, who play on the basis of trust or lack of full information about the rationality of other players
The State
• Historically, the most common ways in which massive free rider problems have been solved have almost invariably involved the state as the agency with a monopoly of violence
• States can address a vast range of prisoner’s dilemma problems from the most fundamental, to several that are recognized by neoclassical economics, to many that take us into the domain of modern debates on the appropriate role of the state in developing countries
• But states can only solve prisoner’s dilemma problems if they themselves have the capacity to enforce. This creation of this capacity can itself represent a political prisoner’s dilemma because some people have to take the initiative from which many others will benefit
The Emergence of Cooperation: Coercion as a Solution of the Prisoner’s Dilemma
The Hobbesian solution to the prisoner’s dilemma problem is external enforcement: the strong state or Leviathan. This is strictly not a game theory solution because it is no longer a non-cooperative game when there is external enforcement, but the non-cooperative game shows why external enforcement may be both necessary and desirable in some contexts.
Problems for the strong state solution:
i) Where does the strong state come from? The creation of a strong state is itself a collective action problem.
ii) What happens if the strong state is rapacious? But why will it be?
Stupidity of leadership, time preference or inability to enforce (weakness of apparently strong state?)
Hobbes and the Leviathan
• Adam Smith writing on the free market in England in the 18th century could never have written if England had remained in the civil war that raged in the 17th century
• Thomas Hobbes was the author who wrote in the 17th century and his concern was how to end the “war of all against all” when life was brutish and short and no economic development was possible
• His solution was the Leviathan, the strong state that could impose order by force, enabling men and women to be free
• This paradoxical relationship between force and freedom is still not adequately recognized in modern political economy and philosophy, where external force and freedom are seen as mutual exclusives that have to reach a compromise or balance
Hobbes’ prisoner’s dilemma
A
B
Live in peace
Pre-emptive strike
8,8
1,1
0,10
10,0
Live in peace
Pre-emptive strike
Other prisoner’s dilemma problems for the state
• Hobbes’ problem of creating a public order is the most fundamental prisoner’s dilemma affecting a society. This public order is unlikely to come about without significant violence capacity on the part of the state
• This explains why state formation is often a process of great violence, but it also often fails if there are distributive issues as well as free riding issues involved
• Moreover, creating public order is only one of many nested prisoner’s dilemma problems
• Once basic order exists, the state has to provide a range of public goods that need to be financed by tax
• This too is prisoner’s dilemma problem that needs to be solved through a combination of force based on democratic requirements for public goods
The taxation prisoner’s dilemma
A
B
Pay taxes required for public goods
Pay less or no taxes
8,8
1,1
0,10
10,0
Pay taxes required for public goods
Pay less or no taxes
This prisoner’s dilemma problem results in low tax collection and poor infrastructure in developing countries
The state is involved even when the solution is largely non-state
• All collective effort involves team work resulting in free rider problems (as in Alchian and Demsetz). Systems of enforcement can make everyone better off but may be resisted because of distributive inequalities (which we discuss later). The emergence of cooperation requires some minimum ‘rules of the game’ to be enforced by the state
• The overuse of commons (tragedy of the commons) is a prisoner’s dilemma problem that can be addressed using the creation of property rights that imposes an enforceable solution
• While these solutions appear to be decentralized solutions, they will only work if behind the firm of private property rights is a strong state that can enforce these rights
Repeated Games and Backward Induction
It may appear that cooperation is more likely to emerge in repeated games, but this is not the case in games with a definite end. Backward Induction shows that in such games, cooperation will not take place if all players are strictly rational and there is common knowledge of the structure of the game.
The game starts with player 1. Each player has the option of choosing “continue” or “stop”. If he chooses Continue, he loses $1 and his opponent gains $2. The game stops as soon as one of the players chooses Stop or if either reaches a wealth of $100.
C
S
C
S S
C
S
C1 2 1 2
1,1
0,3
2,2
98,101
100,100
3,3
1,4
4,4
2,5
99,99
97,100
98,98
96,99
Sub-game Perfect Nash Equilibrium
In games played over time, the strategic form of a game can be misleading in identifying the Nash equilibria which can be sustained over timeSubgame Perfect Nash Equilibrium (SPNE) is a Nash which does not involve any incredible threat: that is the SPNE has to be a Nash for the whole game and for any part of the game still remaining
The strategic form of the game may now be misleading Incumbent
Enter
StayOut
Collude Fight
Entrant
0,100
40,50
0,100
-10,0
The strategic form suggests two Nash equilibria but if the entrant enters, the Fight strategy is no longer rational for the incumbent
Sub-game Perfect Nash Equilibrium
I
E
(0,100)
(-10,0) (40,50)
The extensive form shows there is only one SPNE, that is Enter, Collude
Sub-game Perfect Nash Equilibrium
Selten (1978) in his Chainstore Paradox showed that even if an entry deterrence game is played a large number of times (but finite), the outcome will be the same: there will be no deterrence due to threats to fight
Suppose the incumbent has stores in 20 towns. This time the potential loss from allowing entry is much greater but will the incumbent fight if the payoffs in each market are as shown
The incumbent will not fight in the 20th market if 19 markets have already been invaded since then we are in the one-shot game discussed above
If so, the incumbent will not fight in the 19th market because there is no point in building a reputation for ferocity. And so on down to the first market.
Once again there is only 1 SPNE: Enter, Collude in all 20 markets
Some Definitions of information sets
In a game of perfect information each information set is a single node. This is the finest information partition. Otherwise the game (as in the coordination game) is a game of imperfect information
A game of certainty has no moves by Nature after any player moves. Otherwise it is a game of uncertainty.
In a game of symmetric information, a player's information set at each node where s/he chooses an action and at an end node has at least the same elements as the information set of every other player. Otherwise the game is of asymmetric information
In a game of incomplete information, Nature moves first and is unobserved by at least one player. Otherwise the game is one of complete information. Games of incomplete information have become extremely important in economic modelling. This is because they can be used to model situations where one or all the players are unaware of at the beginning of the game, of the type of game or the type of opponents they are going to be playing.
Incumbents can win in finite games of incomplete information
The entrant has to have some doubt about the kind of incumbent he is fighting (this is what makes the game one of incomplete information)
The incumbent then has the incentive to support the subjective probability p which the entrant may hold that the incumbent is a fighter by actually fighting
The entrant on the other hand has a subjective probability p but knows that the incumbent has an incentive to influence this by fighting! His problem is to decide whether to call the bluff or decide that the process of calling the bluff is too expensive
This game can be modelled within the restrictions of game theory by assuming two types of incumbent, one of whom has a high payoff from fighting (either because the incumbent enjoys fighting or because the incumbent enjoys winning in the end so fighting has positive value)
Entrants can win in finite games of incomplete information
Nature
Incumbent isa CrazyFighter:Probability=p
Incumbentis Rational:Probability=1-p
Entrant
Entrant
Cooperate
Fight
Cooperate
Stay Out
Fight
EnterIncumbent
40,50
-10,0
-10,200
40,50
Stay Out
EnterIncumbent
0,100
0,100
The entrants expected return from entry is (1-p)40 + p(-10). The payoff from staying out is 0. The condition for not entering is therefore 40-50p < 0, ie p > 0.8. Fighting in early games can raise this probability so fighting as an entry deterrence can be rational
Kreps and Wilson, Journal of Economic Theory 1982
Cooperation in Finitely Played Prisoner’s Dilemma with Incomplete Information
Even with a small probability that the other player is a cooperative player (such as a Tit-for-Tat player) rational cooperation can emerge in a finite game despite backward induction
Nature
Tit-for-tatplaying B:Probability=p
Rational B:Probability=1-p
A
A
Cooperate
Cooperate
Defect
Defect
B
B
Cooperate
Defect
Cooperate
Cooperate
Defect
Defect
B
B
8,8
1,1
8,8
0,10
10.0
1,1
A will play Cooperate if
8p + 0(1-p) (the payoff playing cooperate)
>
1p + 1(1-p) (the payoff from playing defect)
ie if p>1/8. Kreps and Wilson, Journal of Economic Theory 1982
Morality, Culture and Trust Revisited
The importance of trust and culture can be modelled in the context of a prisoner’s dilemma game by pointing out that greater trust is a higher probability of meeting a cooperative player like a Tit-for-Tat player. In an incomplete information game, greater trust then increases the probability of cooperation emerging.
However, little evidence that societies where successful collective action emerged, such as in the newly industrializing countries, there was greater social trust than in other countries.
Moreover, trust can result in damaging groups forming, such as mafias and clientelist groups, so it may be that some poorly performing countries have higher levels of trust than some better performing ones.
Trust
• There is a growing literature on trust in new institutional economics arguing that the solution to prisoner’s dilemma problems is to generate trust, which is a belief that others in the group are nice and are going to act in a group-regarding way rather than individualistic way
• But how to generate trust?• There is a dangerous slide explaining trust in terms of culture, for
instance by Jean-Philippe Platteau in his analysis of the growth of market economies, using the role of Christianity to create a particular kind of trust
• This goes back earlier to Weber’s analysis of the Protestant ethic as a contributor to the rise of capitalism
• Cannot explain how ‘problematic’ cultures like Confucianism suddenly became dynamic trust-generating cultures
• A new concern with Islam as a problematic culture in the west is likely to take attention away from more significant issues in developing countries
Repeated Games
• Cooperation can also spontaneously emerge in indefinitely repeated games provided
• i) The gains from cooperation are high and the gains from defection not much higher
• ii) The discount rate is low, • which means that the group is secure about the future and is not
concerned to maximize present consumption
• iii) Punishment strategies can be enforced• which means the group cannot be too large
• None of these features apply to many developing country contexts
Cooperation in Indefinitely Played Prisoner’s Dilemmas
In an indefinitely played game, backward induction is not a problem. Now cooperation can emerge because the gains from cooperation can outweigh the incentive to cheat provided there are adequate punishments for cheating.
The emergence of cooperation depends on i) the size of the payoffs from cooperating Πco (8 units in our example) compared to the payoffs from defecting Πde (10 units) and the payoff from punishment Πpu (1 unit).
ii) The time preference expressed in the discount rate r. Payoffs in subsequent periods are discounted at this rate, so that a payoff in period t is worth Πt/(1+r)t-1 now.
iii) An adequate punishment strategy being adopted by at least one of the players.
The Indefinitely Played Prisoner’s Dilemma Game 2
An example of an effective punishment strategy is the so-called Trigger Strategy: “Cooperate on the current move unless the other player defected on the last move. If the other player defected on the last move, Defect forever”. This is a Nash if each player believes the other will play it. If one player plays this strategy, the other can do no better than by playing it.
If the second player defects, his payoff is: ....
1
1
1
110 2
rr
which sums to 10+ (1/(1+r)/{1-1/(1+r)}= 10+1/r, or, Πde + Πpu/r
If the second player cooperates, his payoff is ....
1
8
1
88 2
rr
which sums to 8+8/r or in general Πco+Πco/r
In this example, the payoff from cooperation is higher if 8+8/r > 10+1/r, which is the case as long as r < 3.5 or in general, r<(Πco-Πpu)/(Πde-Πco)
The Indefinitely Played Prisoner’s Dilemma: The Folk Theorem
The Folk Theorem tells us that Trigger,Cooperate is not the only Nash equilibria in this repeated game.
The Folk Theorem says that in an indefinitely repeated Prisoner's Dilemma with low or zero discount rates, any payoff larger than the payoff received if both parties consistently defect, (1,1 in our example) can be supported as a Nash equilibrium.
Suppose A insists that he will play the following strategy: Cooperate twice in a row and Defect the third time as long as B continues to cooperate but if B ever defects, Defect forever. If discount rates are low enough B's best response to this strategy will be to play Cooperate in every round. A clearly gets a good deal because his sequence of payoffs is 8,8,10,8,8,10.... while B only gets 8,8,0,8,8,0.. but B accepts this because it is better than 1,1,1….
The Folk Theorem Continued
A number of important conclusions follow from the Folk Theorem:
i) Because there are an indefinite number of possible Nash equilibria in the indefinitely played prisoner’s dilemma, any sequence of plays may in principle be observed in a finite part of the game as part of a rational strategy on the part of each player!
This makes it difficult to predict on the basis of game theory what will actually happen at any stage of the game.
ii) With multiple equilibria, there are different distributions of benefits and it is likely that the two players can also try to follow conflict strategies to acquire reputation for being tough and so getting a better Nash from their perspective. This is what can happen in a Chicken Game, so a Repeated Prisoner’s Dilemma game can take on some characteristics of a Chicken Game.
Multi-Person Prisoner’s Dilemma Games
2 Person Prisoner’s dilemma games do not give a full idea of the problem of collective action in a many-person game. The problem is that a multi-person game cannot be represented on a 2-dimensional page. One solution is the diagram proposed by Schelling which looks at the decision problem from the perspective of a single individual.
n
Utility ofn+1 thindividual
0
Cooperate
Defect
Number ofIndividualsCooperating
a
b
c
Multi-Person Prisoner’s Dilemma Games 2
The shape of the Defect curve is given by the technology of producing the public good which the collective action produces. If there are economies of scale in the production of the public good, the D curve will show this in the form of greater than proportionate increases in the utility accruing to the n+1th player as the number of cooperators increases.
Conversely, if there are decreasing returns to scale in the production of the public good, the D curve will become flatter and flatter.
The gap between the D curve and the C curve depends on whether the per capita cost of contributing goes up, down or stays constant as the number of contributors increases.
The break even number of contributors, k, is the number which if they cooperated, would earn at least the payoff earned when cooperation breaks down completely and everyone defects (0)
Multi-Person Prisoner’s Dilemma Games 3
n
Utility ofn+1 thindividual
0Cooperate
Defect
Number ofIndividualsCooperating
a
b
c
k
n
Utility ofn+1 thindividual
0
Cooperate
Defect
Number ofIndividualsCooperating
a
bc
k
If k is small, cooperation is more likely to emerge because the cooperators can suggest a strategy such as “I will cooperate as long as at least k others cooperate, but I will defect forever if the number of cooperators falls below k”.
This may be sufficient to sustain cooperation, provided that the number of cooperators can be monitored, and this is easier when k is small.
Distributive conflict as another problem facing the state
• Every prisoner’s dilemma problem also has embedded within it a distributive conflict because the different players in the game never gain or lose exactly the same amount despite the simplification in our payoffs
• When there is a gain from social cooperation but the gains are very unevenly divided, we have a problem of cooperation with conflict.
• This is just as difficult a problem to solve as a prisoner’s dilemma, and the solution is less clear cut as these games have multiple equilibria
The Chicken Game (Cooperation with Conflict) B
Dove/Worker
Hawk/Capitalist
Dove/Worker Hawk/Capitalist
A
-5,-5
5,5
10,3
3,10
The payoff structure is such that
i) Complementary actions maximize the social payoff
ii) the Distributive implications are unequal
iii) if both players play the soft option each is better off than if the socially desirable outcome happens and they are the weaker party:
There are 2 pure strategy Nash Equilibria with unequal distributive consequences.
The Chicken Game 2
The equilibrium that emerges and the process through which it emerges cannot be explained by game theory.
This process can involve a considerable period of conflict as each side attempts to gain the reputation of being the tougher side.
Knight shows how the distribution of power between the protagonists of a chicken game can determine who is the winner of the game. The more equal the holding power of the agents, the longer the conflict will continue. Paradoxically, this suggests that the more unequal the initial distribution of power, the faster the resolution to the conflict.
Chicken Game 3
Knight uses the following bargaining game:
B
Left Right
Left ΔA, ΔB x, x+εB
A
Right x+εA, x ΔA, ΔB
To get the Chicken game problem, we specify that ΔA, ΔB < x, which
ensures that (Right, Left) and (Left, Right) are the socially desirable strategy combinations, and the ε ensures that each player wants to play Right. ΔA, ΔB are the breakdown values which means the players get at
least this if coordination breaks down.
Chicken Game 4
Knight argues that Resource Asymmetries can help to communicate the intention to fight because this gives one player greater credibility in declaring an intention to fight, because that player is recognized as having lower risk aversion and lower time discount rates.
If ΔA > ΔB, A is deemed to have greater bargaining power because A can
credibly take greater risks and hold out for longer in conflicts. Since B knows this, B may play Left straight away rather than testing A's resolve.
This analysis shows that the resolution of some collective action problems may have little to do with equity or justice.
However, the willingness to fight does not just depend on economic power as represented by breakdown values: it also depends on organizational success and the psychological willingness to fight (based on notions of injustice) that are difficult to pin down in a rational choice model
Implications of the Cooperation with Conflict Game
• The resolution of the basic order problem of creating property rights is not likely to be helped by any of the governance measures that are recommended in contemporary development theory
• The more equal the power of the contestants, the more prolonged the conflict between would-be capitalists, and the higher the social cost (Jack Knight’s analysis of institutions and power)
• The state cannot be invoked to solve a long-lasting conflict game because the power of the state is not independent of society
• In other words, if society is fractured into many factions, the state is also likely to be fractured
• The only possible solution is the reorganization of new political forces through new political organizations (political parties) where ambitious, nationalistic leaders take on the costs and risks of solving the prisoner’s dilemma problem of creating a strong state
Variants of collective action problems and their interrelationship
Player 1
Player 2
C
D
C C C CD D D D
3,3
4,1
1,4
2,2
Prisoner’s dilemma
3,3 2,4
4,2 1,1
Chicken game
4,4 1,3
2,23,1
Assurance game
3,2
1,1
1,1
2,3
Battle of the sexes
Prisoner’s dilemma: Nash is not socially optimal
Chicken: Multiple Nash with the probability of conflict
Assurance: Multiple Nash but social optimum can be achieved through signalling (coordination). Free riding does not happen because the public good is very ‘lumpy’ and without both contributing, it is not satisfactorily provided
BoS: Multiple Nash but here social optimum is coordinated activity unlike Chicken
Player 1
Player 2
C
D
D
Relationship between prisoner’s dilemma and chicken
• We have already seen in a 2-person prisoner’s dilemma that is indefinitely repeated that there are multiple Nash equilibria based on different punishment strategies that give different benefits to the 2 players (Folk Theorem)
• Therefore the emergence of a SPNE requires the punishment strategies of one of the players to be credible, particularly if it gives that player a better distribution
• The emergence of this credibility could therefore be preceded by many rounds of Chicken Game conflict as each player tries to acquire a reputation for being tough
• We will now see a related problem in the n-person prisoner’s dilemma that establishes yet another type of Chicken game being played
• We have seen that in the Schelling diagram, there is usually a critical minimum number of individuals who have to cooperate if any cooperation is to be sustained
Relationship between prisoner’s dilemma and chicken
• We have seen that in the Schelling diagram, there is usually a critical minimum number k of individuals who have to cooperate if any cooperation is to be sustained
• All members of the bigger group (n) want some subgroup k to contribute, but no-one wants to be part of that sub-group (Udehn, Taylor)
• A Chicken Game can break out between an indefinitely large number of potential subgroups where each wants to signal that it will not cooperate but in fact if others really will not cooperate, it will cooperate out of self-interest: the bigger the group n the more complex this Chicken game can become
n
Utility ofn+1 thindividual
0
Cooperate
Defect
Number ofIndividualsCooperating
a
b
c
k
Relationship between prisoner’s dilemma and chicken
• The multi-person prisoner’s dilemma can also have exactly the same multiple Nash problem of the 2-person game because cooperation within the cooperating group of k individuals can be sustained on many different punishment strategies within the cooperating group (Trigger and its extensions)
• Therefore Chicken games can also emerge within this group as some cooperators attempt to get a better deal by signalling they will only partially cooperate
• However, these strategies are even more unlikely to be sustained as the observation requirements when there are a large number of players become implausible
• Indeed, the transaction cost requirements of sustaining punishment strategies in any n-person prisoner’s dilemma is a sophisticated reason why cooperation is unlikely to be sustained in large groups that goes beyond Olson’s simple arguments
Prisoner’s dilemmas and assurance games
• The most plausible type of strategy that may be sustained in an n-person prisoner’s dilemma is one that says I will cooperate if at least a proportion of others cooperate, but if that proportion falls below k, I will never cooperate again
• Clearly, since the cooperators and non-cooperators are a changing mix, this type of strategy is very unstable and can collapse for stochastic reasons
• This too is another reason supporting Olson’s simplistic observation based on his unrealistic model of privileged individuals providing public goods
• Some authors (Amartya Sen) have argued that many collective action problems are actually games of assurance and can be resolved by signalling
• These arguments are typically not convincing in large number games: it is plausible that a 2 or 3 person game of public good provision can be an assurance game but it is unlikely that any public good is so lumpy that it will remain an assurance game as numbers of potential contributors increase: yet more support for Olson
If large number cooperation is so unlikely how does it happen?
• Game theory tells us that rational individuals are very unlikely to cooperate in large number situations where free riding is likely to pay off
• Yet there are many everyday instances of large number cooperation that defies game theory logic: the voting paradox
• Clearly the rational choice framework on its own is a dangerous tool if taken too far because it ignores many irrational elements that can sustain both useful and damaging collective action
• Modelling collective action should therefore be open to these other factors: Elster’s Mixed Motives (Ch.5 of The Cement of Society)
• He argues that it is impossible to explain collective action without taking into account a variety of motivations that may sustain collective action at different stages of its evolution
Elster’s Mixed Motives Approach• It is not very illuminating to simply say that collective action depends on
‘norms’ or culture. Elster distinguishes between 5 different types of motives/individuals to better explain collective action:
• Everyday Kantians: These are people who will cooperate under most circumstances because they follow the Kantian ‘categorical imperative’ that says some categories of laws (derived from pure reason) always have to be followed.
• The simplest way to understand this is that Kantians follow rules derived from the precept ‘do as you would be done by’.
• Since this is not strictly rational because individuals could harm themselves by following this rule, Elster provides a psychological explanation of why a few people in any society are likely to be everyday Kantians (and we only need a small number to start off collective action as catalysts)
• His explanation is that some people search for magical causality where they psychologically choose an outcome that they think may make it more likely that a cause that they want to believe in is actually operating
Elster’s Mixed Motives Approach• Quattrone and Tversky’s experiment: subjects were told that some
people had a different kind of heart that resulted in longer life and this could be determined by their greater or lower tolerance to cold water after exercise
• When tested later, a significant number endured cold water for a longer or shorter time (depending of what they had been told) compared to before they were exposed to the information
• It appears that by choosing an outcome, individuals were trying to improve the chance that they were of the lucky type
• Alternatively, they wanted to believe that they were lucky and this affected their choices
• This is also similar to the religious problem known as Newcomb’s problem: why should religious people who believe in predestination go to church, if god has already selected or not selected them as the ‘elect’ or good people
• If they were rational, going or not going to church would make no sense, because God must already have decided their future. But going to church could reveal that god had selected them as part of the elect, so this ‘decision’ can be explained in terms of a search for magical causality
Elster’s Mixed Motives Approach• This type of thinking can be transposed to the prisoner’s
dilemma problem • If two people are sufficiently alike, they may reason that if I
cooperate he will cooperate too, because being like me, if I cooperate that increases the chance that this is how rational people think, so he will cooperate too
• The similarity with the previous examples is that here too, the behaviour of the Kantian only makes sense in terms of a magical causality: the individual who we still believe to be rational is operating on a faulty logic that their decision can change some prior likelihood (in this case that other individuals will behave like them)
• Of course, Elster is trying to make sense of Kantians in terms of rationality, it may be that this is a meaningless approach and that we cannot analyse Kantians (or religious people) in this way
• In any case, his approach only requires a small number of Kantians in society to act as catalysts
Elster’s Mixed Motives Approach• Utilitarians: They operate according to the moral norms of
utilitarianism that says that actions should be chosen in terms of whether they enhance total social utility.
• Thus, utilitarians will cooperate if the average payoff from cooperating is rising. These people too are not entirely rational in terms of rational choice, because they are going to be better off by defecting
• But unlike Kantians, utilitarians will not initiate cooperation if no-one is cooperating
• Whereas a Kantian will cooperate whatever happens to their own utility or that of society, the utilitarian will cooperate only if their cooperation raises the average utility in society
• This is also a moral position, but it is a less strict position than that of the Kantian
• Once cooperation begins, the presence of utilitarians in society can increase the number of cooperators
Elster’s Mixed Motives Approach• Individuals motivated by Norms of Fairness These individuals
cooperate when the number of cooperators exceeds some threshold. This kind of cooperator may be motivated by feelings of shame: if everyone or a large number are cooperating, non-cooperation hurts their sense of self-worth
• However different individuals may have different thresholds after which their norm of fairness kicks in. Some individuals may not start cooperating till everyone else cooperates, others may start cooperating at lower thresholds
• Like Kantians, fairness motivation can not only harm the individual following it, but also society
• This is possible if cooperation at very low (Kantian) or very high (fairness) levels results in declining collective benefits
• Examples would be resistance to occupation if only a few people resist or clearing up after a party, if everyone insists on helping
Elster’s Mixed Motives Approach• Participationists These are people who get a benefit from
participating. Here the issue is not a moral norm, but utility from participating, so these people are more strictly rational, except they have some special sources of utility
• Participationists can either be elite, who get pleasure if very few people cooperate, or mass, who participate when many participate. They will cooperate at different stages of collective action. Finally we have
• Rational Individuals They will never cooperate in a 1-shot game but a) the numbers of pure rationalists may not be large enough to prevent a significant amount of cooperation, and b) repeated games and punishment strategies may induce many of them to cooperate in the end
• In Elster’s mixed motives approach, collective action is set off by Kantians, if the average payoff rises enough they are joined by utilitarians and perhaps elite participationists. Eventually people motivated by norms of fairness and mass participationists join and collective action can become sustainable with some limited punishment strategies or enforcement for the purely rational
State and the Problem of Enforcement
Time Horizon andEnforcement Capacity
of the State(success in enforcing
resolutions toPrisoner’s Dilemma games)
Emergence of aViable Economy
Factional Organizationof Society
(intensity of Cooperation with Conflict Games)
The Collective Action Challenges of Development
• The necessary interventions can be broken down in problems of enforcement and the distributive conflicts that can prevent enforcement
• Enforcement cannot be discussed without clarifying
• a) the interventions that are necessary for accelerating transformation in developing countries
• b) the distributive conflicts that are preventing their implementation