Introduction

• In the previous Unit we model how individuals (acting own their own as one decision maker) choose the best possible outcome given constraints
• Sometimes the outcome depends on the actions of others as well as your own actions (more than one decision maker)
• In this Unit, we will use the tools of game theory to model social interactions⁠, in which the decisions of individuals affect other people as well as themselves.
• We will examine when and why social dilemmas arise, and how people can sometimes solve them.

Overview

• Our economy is shaped by millions of direct and indirect interactions among people.
• These social interactions offer opportunities for mutual gains for instance, gains from trade
• But conflicts often arise over how these gains should be distributed.
• The individual pursuit of self-interest may lead to socially beneficial outcomes, as indicated in the idea of an ‘invisible hand’
• But there are interactions, called social dilemmas, in which people would do better by cooperating rather than acting individually. When they don’t cooperate (but act in their self interest) the outcome is worse for everybody, including themselves.
• These social dilemmas occur when people do not take into account the effects of their actions on others, called external effects
• And they give rise to the problem of free riding, where people benefit from the contributions of others to a cooperative project without contributing themselves.
• Public goods are an example of a cooperative project where free riding means that the market will not generate an efficient outcome.

The tragedy of the commons

Picture a pasture open to all.

Each herdsman seeks to maximize his gain and will try to keep as many cattle as possible on the commons.

He asks, ‘What is the benefit to me of adding one more animal?’

The benefit can be divided into

1. The positive component: the herdsman receives all of the proceeds from the sale of the additional animal.
2. The negative component: the effects of overgrazing are shared by all of the herdsman, so the negative impact for any decision-making herdsman is only a fraction of the total negative effect.

The only sensible course for him to pursue is to add another animal to his herd. And another….

But this is the conclusion reached by each and every herdsman sharing a commons. Therein is the tragedy. Ruin is the destination towards which all men rush, each pursuing his own best interest.

Freedom in the commons brings ruin to all.

Paraphrased from Hardin (1968)

The tragedy of the commons relevant to common-pool resources

• Common-pool resources
• Resources that are shared, not owned by anyone
• Resources of this type (e.g. the earth’s atmosphere or fish stocks) are easily overexploited unless we control access in some way
• If you reduce your carbon footprint, or limit the number of fish you catch, you will bear the costs, while others will enjoy the benefits
• Also relevant is the fallacy of composition –
• The fallacy of composition refers to the logically untenable position that what is true for a member of a group must necessarily also be true for the group as a whole
• e.g. if you stands up in the theatre you can see better, but it is not the case if everyone stands up

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Social Dilemmas

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• A situation in which actions taken independently by self-interested individuals result in a socially suboptimal outcome e.g. the tragedy of the commons, traffic jams, climate change
• Social dilemmas occur frequently and diminish the quality of our lives and the lives of others.
• One of the tasks of public policies is to address social dilemmas.

Resolving social dilemmas

• Altruistic preferences
• A person with these preferences cares about the implications for other people. Given the chance he would prefer to help some other person, even if it cost something to do so.
• There are many examples of willingness, or even desire, to help others even at a cost to oneself (e.g. during wars, natural disasters – see recent book by Rutger Bregman Humankind: A Hopeful History)
• Community institutions
• Local communities can create their own institutions to regulate behaviour
• An example: Customary rules designed by farmers in Valencia, Spain, and the Tribunal de las Aguas (Water Court) to resolve conflicts
• Another example: taxi marshals?
• Public policies
• Taxes and government spending
• Laws and Regulation (e.g. Fishing quotas)

Game Theory – modelling how people interact

• Social interactions⁠ are situations in which there are two or more people, and the actions taken by each person affect both their own outcome and other people’s outcomes.
• Defining terminology:
• When people are engaged in a social interaction and are aware of the ways that their actions affect others, and vice versa, we call this a strategic interaction⁠.
• A strategy⁠ is defined as an action (or action plan) that a person may choose while being aware of the mutual dependence of the outcomes on their own and others’ actions.
• Models of strategic interactions are described as games⁠.
• Game theory⁠ is a set of models of strategic interactions. It is widely used in economics and elsewhere in the social sciences.

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Using game theory to model farmers’ choices of crops

There are two famers Anil and Bala

They each need to choose whether to plant Rice or Cassava

Assumptions

They act independently and do not meet to plan, they decide simultaneously, and they know the outcome or payoff in each situation

Their land can grow either crop, but they each choose to produce only one crop

Anil’s land can produce rice and cassava equally well.

Bala’s land is better for producing rice, and no suitable for producing cassava

If they both produce the same crop supply of that crop will be greater (called a market glut) and prices will be lower

If they each produce a different crop – supply of each crop will be less (there will not be a market glut) and prices of both crops will be higher

Results

The figures show he results of the game or crop choices by Anil and Bala

The top figure shows the results in words

The bottom figure shoes the payoff to Anil and Bala in Numbers

Anil is the row player and Bala is the column player

Pay offs

Using game theory to model farmers’ choices of crops

There are for four possible outcomes, for example:

Top left shows an outcome where Anil produces rice and Bala produces rice, therefore the price of rice will be lower as both or producing same crop (glut), it is better for Bala to produce rice than cassava, for Anil the production of both crops is suitable

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Bottom left shows an outcome where Anil produces cassava and Bala produces rice, therefore, the price of rice and cassava will be higher (no glut) and it is better for Bala to produce rice than cassava, for Anil the production of both crops is suitable

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The lower figure shows the payoffs (or incomes) they would receive if each of the hypothetical row and column actions are taken

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Payoff top left – Anil gets 4 and Bala gets 4 (as price of rice is lower due to glut, and it is better for Bala to produce rice than cassava when Anil produces rice, for Bala 4>3)

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Payoff bottom left – Anil gets 6 and Bala gets 6 (as price of rice and price of cassava is higher as there is no glut, Anil’s land is suitable for producing cassava and Bala is producing rice for which his land is suitable)(if Bala produces cassava when Anil produces cassava anil would get 5 (due to suitable crop, but glut) and Bala would get 2 (due to unsuitable crop and glut)

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Pay offs

Outcomes

Best Response

• Best response⁠ is the strategy that will lead to a player’s most preferred outcome, given the strategies that the other players select
• The pay-off matrix reveals that there is one pair of strategies which are best responses to each other: Anil chooses Cassava and Bala chooses Rice.
• We call this pair of strategies an equilibrium⁠ of the game.
• An equilibrium is a self-perpetuating situation (where no player has an incentive to play an alternative strategy)
• In this case, Anil choosing Cassava and Bala choosing Rice is an equilibrium, because neither of them would want to change their decision after seeing what the other player chose.

If B chooses rice, A chooses cassava (6>4)

If B chooses cassava, A chooses cassava (6>5)

If A chooses rice, B chooses rice (4>3)

If A chooses cassava, B chooses rise (6>2)

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Nash Equilibrium

• In game theory, a Nash equilibrium occurs when a set of strategies is adopted in which each player plays a best response to the strategies of the other players 
• In this game, A chooses cassava and B chooses rice is a Nash equilibrium
• Can we predict that Anil and Bala will play their Nash equilibrium strategies?
• Thinking about the decision from Anil’s point of view suggests that he might not be sure what to do, because his best response depends on Bala’s decision (he wants to make the opposite choice from Bala).
• However, Bala’s decision is easier (he has a dominant strategy): whatever Anil does, Bala gets a higher pay-off from choosing Rice. So we would expect him to choose Rice.
• Then, if Anil thinks about Bala’s decision, he will also expect Bala to choose Rice. This simplifies the problem for Anil: his best response to Rice is Cassava.
• Usually, if we find that a game has just one Nash equilibrium, it is the most plausible or most likely outcome—although we may feel more confident about this in some games than others (we will see later that some games can also have more than one Nash equilibrium)

If B chooses rice, A chooses cassava (6>4)

If B chooses cassava, A chooses rice (6>5)

If A chooses rice, B chooses rice (4>3)

If A chooses cassava, B chooses rice (6>2)

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Invisible hand game

• In the Nash equilibrium, both Anil and Bala specialize in one crop, so market gluts are avoided.
• Moreover, both crops are produced on land well suited for growing them.
• Simply pursuing their self-interest—that is, choosing the strategy giving them the highest pay-off—results in an outcome with the highest possible pay-off for each player and thus the highest outcome (6 + 6 is highest total pay-off for society).
• This is an example of an invisible hand game as it is also the outcome that each would have chosen if they had a way of coordinating their decisions.
• Although they independently pursued their self-interest, they were guided by market prices to an outcome that was in their best interests and society’s.

If B chooses rice, A chooses cassava (6>4)

If B chooses cassava, A chooses rice (6>5)

If A chooses rice, B chooses rice (4>3)

If A chooses cassava, B chooses rice (6>2)

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Dominant strategy equilibrium

• If as indicated in a different payoff matrix Bala’s land is better for growing rice; and Anil’s land is better for cassava (no matter what Bala does)
• The Nash equilibrium is the same as before: Anil chooses Cassava and Bala chooses Rice.
• But because Cassava is a dominant strategy for Anil and Rice is a dominant strategy for Bala, we can be especially confident in predicting this outcome.
• We say that (Cassava, Rice) is not only a Nash equilibrium but also a dominant strategy equilibrium⁠
• This is also an example of an invisible hand game where the pursuit of self -interest (the dominant strategy leads to the best outcome for both A and B and for society at payoff 6+6)

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- a dominant strategy equilibrium involves a player's best move regardless of the opponent's actions,

- a Nash equilibrium is a player's best move given the opponent's strategy

Invisible hand game

• Such a game is called an invisible hand game⁠, because it reflects Adam Smith’s idea that forces that are not explicit (‘invisible’) can guide the players to the outcome that is best for both of them.
• An invisible hand game has the property that players acting independently in their own self-interest reach an equilibrium that is in the joint interest of the players involved.
• This is different from a prisoners’ dilemma game where the pursuit of self interest leads to sub-optimal outcomes and socially optimal outcomes are rather achieved through co-operation between the players (rather than simply the pursuit of self interest)

Prisoners’ dilemma game

• Anil and Bala are now facing another problem - how to deal with pest insects that destroy the crops they cultivate in their adjacent fields.
• Each has two feasible strategies:
• use an inexpensive chemical called Toxic Tide, it kills every insect for miles around, it also leaks into the water supply that they both use
• use integrated pest control (IPC), in which beneficial insects are introduced to the farm, to eat the pest insects
• If just one of them chooses Toxic Tide, the damage is quite limited. If they both use it, water contamination becomes a serious problem, and they need to buy a costly filtering system.
• Anil and Bala know that their pay-off (the income from selling their crops, minus the costs of their pest control method and of any water filtration required), will depend not only on their own decision, but also on the other’s choice

Prisoners’ dilemma game

• Anil’s best responses are:
• if Bala chooses IPC: Anil chooses Toxic Tide (cheap eradication of pests, little water contamination) (4>3)
• if Bala chooses Toxic Tide:  Anil chooses Toxic Tide (IPC costs more and cannot work since Bala’s chemicals will kill beneficial pests) (2>1)
• Bala’s best responses are:
• if Anil chooses IPC, Bala chooses Toxic Tide (4>3)
• If Anil chooses Toxic Tide, Bala chooses Toxic Tide (2>1)
• Dominant strategy equilibrium is that both Anil and Bala will use Toxic Tide – so payoff for Anil is 2 and payoff for Bala is 2
• This is not the best outcome for Anil and Bala if they rather trust each other and cooperate and both use IPC, the payoff for Anil is 3 and payoff for Bala is 3
• Cooperation would be more socially optimal as 3+3>2+2
• 2+2 is a Nash equilibrium as no player can improve their outcome by changing their strategy, assuming the other player’s strategy stays the same

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Why is it called the prisoners’ dilemma?

• Textbook example - where the pursuit of self-interest leads to results that are not optimal.
• Optimal outcome requires cooperation between the two players but co-operation is difficult and requires trust
• Two people caught with drugs: the sentence is one year.
• Both are also charged with murder, which carries a life sentence, but the police don’t have evidence and therefore need a confession.
• If both confess, the police show leniency, and each gets a sentence of ten years
• If only one confesses and the other keeps quiet the confessor is rewarded for providing evidence against the other and goes free (drug charges are dropped) and the other gets a 30-year life sentence.
• If both keep quiet, they both get one year for the drugs charges (this is optimal for the prisoners as total time in jail is 1 year plus 1 year = 2 years)
• But the police keep them apart so trust and cooperation are difficult to achieve.

The prisoners’ dilemma

• For Prisoner A
• If B confesses, then Prisoner A’s best strategy is to confess and get 10 years (if he keeps quiet he get 30 years)
• If B Keeps quiet, then Prisoner A’s best strategy is to confess and get 0 years rather than 1 year if he keep’s quiet

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• For Prisoner B
• If A confesses, then Prisoner B’s best strategy is to confess and get 10 years (if he keeps quiet he get 30 years)
• If A Keeps quiet, then Prisoner B’s best strategy is to confess and get 0 years rather than 1 year if he quiet

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• Therefore, the dominant strategy for both is to confess so they both get 10 years (-10, -10) (total jail time 20 years)
• But the optimum strategy is for both A and B to keep quiet and get one year (-1, -1) (total jail time is 2 years)
• The risk of keeping quiet is that if a prisoner keeps quite and the other prisoner confesses then the prisoner who keeps quiet will receive 30 years in prison
• In a prisoner’s dilemma there is something to gain when everybody cooperates (i.e. keeps quiet) , but each person can loose if they cooperate and the other defects (i.e. confesses)
• The socially optimal outcome is not achieved when people choosing to confess
• They would achieve a better outcome by both keeping quiet.

Games with two Nash equilibria – coordination game

• Anil’s land is better for growing cassava, and Bala’s for rice
• But in this payoff matrix if the two farmers produce the same crop, there is such a large fall in price that it is better for each to specialize, even in the crop they are less suited to grow
• Therefore each farmer will always prefer to produce a different crop to the other farmer but they do not communicate
• This is a coordination game where there is more than one Nash equilibrium, and if the farmers choose their actions independently (and do not communicate), then the players can get ‘stuck’ in an equilibrium in which all players are worse off than they would be at the other equilibrium

6+6 is clearly better than 4+4 but farmers may choose 4+4 (e.g. if Anil starts with growing Rice, then it is in Bala’s interest to grow Cassava (4>3) even though it would be better for Anil to grow Cassava and Bala to grow rice)

Note: there is no conflict of interest between Anil and Bala as they both receive the same outcomes as eachother at each Nash equilibrium

Games with two Nash equilibria – coordination game with conflict of interest

• A conflict of interest occurs in a coordination game if players in the game would prefer different Nash equilibria
• From the payoff matrix software engineers Astrid and Bettina
• Would both do better if they work in the same language
• Astrid does better if that language is Java, while C++ is better for Bettina
• Their total pay-off is higher if they choose C++
• There are two Nash equilibria – one where both choose Java, the other where both choose C++

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But which Nash equilibrium would they choose? Astrid obviously prefers that they both play Java while Bettina prefers that they both play C++

But the total pay-off from the project (3+6) is higher if both choose C++. If they could agree that both would use C++, perhaps they could also agree to split the proceeds in a way that would make both of them content with the outcome

Deciding whether outcomes are Pareto efficient

• The Pareto criterion states that an allocation is better if it makes at least one person better off and no one worse off
• An allocation that is not Pareto-dominated by any other allocation is called Pareto efficient⁠. 
• If an allocation is Pareto efficient, then there is no alternative allocation in which at least one party would be better off and nobody worse off
• For example, Anil playing IPC and Bala playing Toxic Tide is Pareto efficient (moving to any other allocation would make at least one player worse off that is Bala would have less than 4 at any other allocation)
• Although we (and Anil) may think (I, T) is unfair: Anil’s pay-off is 1, while Bala gets 4, it is nonetheless Pareto efficient�

Comparing various outcomes

• Comparing (I, I) and (T, T): The allocation at (I, I) gives Anil 3 and Bala 3 as compared to (T, T) which gives Anil 2 and Bala 2, so (I, I) Pareto-dominates (T, T) – in moving from T, T both can be made better off and no one is made worse off (so even though the Nash equilibrium in the prisoner’s dilemma game is (T, T), (I, I) would be a Pareto improvement if Anil and Bala were able to cooperate)
• Comparing (T, T) and (T, I): If Anil uses Toxic Tide and Bala IPC, then Anil is better off but Bala is worse off than when both use Toxic Tide. The Pareto criterion cannot say which of these allocations is better.

We need to be careful with the concept of Pareto efficiency for two reasons

• Firstly, it may not tell us much about what is best:
• There is often more than one Pareto-efficient allocation: in the pest control game, there are three.
• The Pareto criterion does not tell us which Pareto-efficient allocation is best: it does not rank (I, I), (I, T), and (T, I).
• Even by the Pareto criterion, a Pareto-efficient allocation is not always better than a Pareto-inefficient one: we know that (T, I) is Pareto efficient, and (T, T) is not. But if you compare the two, (T, I) does not Pareto-dominate (T, T)
• Secondly, there may be other criteria that matter as much, or more. In particular, we may want to assess both fairness⁠ and Pareto efficiency when we evaluate outcomes.

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Public goods (example of paying for irrigation)

• Four small farmers share the benefits of a common irrigation system
• Each farmer decides whether to contribute to maintenance of the common irrigation system
• Contributing has a personal cost of $10 each.
• The $10 is the minimum payment to fix up leaks, remove weeds etc.
• For each $10 contributed the improved flow of waters benefits all four farmers, due to improve crop yield, the value of that benefit is $8 due to increased crop yield.
• i.e for every $10 contributed, each farmer receives a personal benefit of $8.
• For each farmer that joins the value of the irrigation systems grows for this farmer and other farmers by $8 so if 1 farmer joins the payoff for all farmers is $8, if 2 farmers join it is $16, etc.
• The public benefit of more farmers contributing to the irrigation system is not in doubt, the problem is that for each individual farmer the cost of contributing ($10) is higher than his or her personal benefit ($8) by $2
• So, each farmer has an incentive not to contribute (if not compelled to contribute) and to be a free rider (and gain access to a public good without contributing)
• Public Good: a good for which use by one person does not reduce its availability to others (also called a non-rival good)

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Public goods game

• Let’s look at it from the perspective of one farmer: Kim. She has two feasible strategies: contribute (i.e. cooperate) or don’t contribute (i.e. defect)
• For example, if two of the other farmers contribute, Kim will receive a benefit of $8 from each of their contributions:
• Option 1 if Kim makes no contribution herself, her total payoff is $16.
• Option 2: If Kim decides to contribute, she will receive an additional benefit of $8 but she will incur a cost of $10, so her total payoff is $14

Calculation for Kim if two other farmers are contributing

Kim’s payoffs in the public goods game

How to overcome the free-rider problem

• Kim will have an incentive not to contribute to the the public good and be a free rider e.g. gaining a payoff of $16 instead of $14
• It is Kim’s dominant strategy not to contribute, as a result fewer resources overall will be invested in the public good irrigation scheme.
• It is a prisoner’s dilemma scenario - where the pursuit of self-interest leads to less than optimal outcomes
• If the farmers care only about their own monetary payoff, there is a dominant strategy equilibrium in which no one contributes and their payoffs are all zero.
• Everyone would benefit if everyone cooperated, but irrespective of what others do, each farmer personally does better by free riding on the others
• Enforcement could help to solve the free rider problem : Economist Elinor Ostrom showed that sustainable use of common property may be enforced by actions that clearly deviated from the hypothesis of material self-interest. In particular, individuals would willingly bear considerable costs to punish violators of rules or norms. In this way the free-rider problem and the tragedy of the commons may be overcome through self-governing community organisation.
• Altruism could also help to solve the free rider problem: if Kim cared about the other farmers, she might be willing to contribute to the irrigation project. But if large numbers of people are involved in a public goods game, it is less likely that altruism will be sufficient to sustain a mutually beneficial outcome.

Summary of Types of Games analysed so far

• Crop choice game (Invisible hand) the dominant strategy of all players results in mutually beneficial outcomes
• Pesticide game (Prisoner’s dilemma): The dominant strategy equilibrium results in a sub-optimal outcome for all players
• Irrigation game (Public goods): same result as the prisoner’s dilemma (but with more than two players) in that the incentive for individuals to be free riders leads to sub-optimal outcome for all players

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Social Preference for Altruism

• In real-world examples and experiments, people often play the cooperative strategy in prisoners’ dilemma games—rather than choosing to defect, the dominant strategy for self-interested players. One possible explanation is altruism
• What is altruism?
• Altruism is a social preference in which an individual’s utility is increased by benefits to others
• Other social preferences are inequality aversion (a preference for more equal outcomes); and spite and envy—in which cases, benefits to others may reduce the individual’s utility
• So far, we have assumed self-interest in our game-theoretic models, with each agent’s utility given by their own pay-off
• But people generally do care about what happens to others. When people have social preferences⁠, their utility depends not only on what they obtain for themselves, but also on things that affect the wellbeing of other people

Altruism and the winning of the lottery

• Zoë is given some tickets for the national lottery, and one of them wins a prize of £200.
• Will she decide to keep all the money for herself, or share some of it with her flatmate, Yvonne?
• Her decision depends on how much she cares about Yvonne: that is, on whether Zoë has altruistic or self-interested preferences in this situation
• This is not game theory as it is about the decision making of one person Zoë
• Zoë’s problem is how to allocate her ‘budget’ of £200 between two ‘goods’: her own share, and if she is altruistic, Yvonne’s share
• Zoe’s Indifference curves have different shapes if she has altruistic preferences or if she has self-interested preferences

Altruistic vs self-interested preferences

• When Zoë has altruistic preferences (which are subjective) her indifference curves are non-linear and slope downwards (with a negative slope)
• For example, along the lowest indifference curve Zoë has the same utility at near 200 for Zoë and near 0 for Yvonne as at near 160 for Zoë and at near 20 for Yvonne (Zoë gains some utility for herself from Yvonne receiving some money)
• When Zoë has a relatively large amount (near 200) she maintains equal utility by giving up a relatively large amount for herself on the horizontal axis for Yvonne to receive a particular amount on the vertical axis
• When Zoë has a relatively smaller amount (like 80) she will maintain equal utility by giving up a relatively small amount for herself on the horizontal axis for Yvonne to receive a particular amount on the vertical axis
• Zoë will always prefer to be on the highest possible/feasible indifference curve (furthest from the origin) where her combined utility for herself and Yvonne is highest

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Altruistic vs self-interested preferences

• When Zoë has purely self-interested preferences (which are subjective) then her indifference curves are vertical
• Along the vertical indifference curve say at 80, Zoë only gains utility from the fact that she has 80 and her utility is not affected by any amount that Yvonne might have (from 0 to 240) on the vertical axis
• Zoë’s utility increases as her amount increases along the horizontal axis from 80 to 140 to 200 as is indicated by the rightward shift in her indifference cure away from the origin

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• If Zoë is more altruistic, then her indifference curves would be flatter;
• if she were more self-interested, they would be steeper
• If she is purely self-interest they would be vertical

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Altruism does not mean that Zoë cares as much about Yvonne as herself. Altruism means that Zoë gets some utility from seeing Yvonne have some share of the lottery money.

Zoë’s decision depends on intersection of highest indifference curve and feasible frontier

• Feasible frontier shows that Zoë could receive 200 or Yvonne could receive 200 and any points on the frontier joining these two points
• This is an objective relationship depending on how much money is available from the lottery to be shared (or not shared)
• It is a zero sum game (any gain from Yvonne results in an equal a loss for Zoë)
• if Zoë keeps 200 Yvonne gets 0
• if Zoë keeps 190 (-10) Yvonne gets 10 (+10) [-10+10 = 0 therefore zero sum game)
• if Zoë keeps 160 (-40) Yvonne gets 40 (+40) [-40+40 = 0]
• if Zoë keeps 80 (-120) Yvonne gets 120 (+120) [-120+120 = 0]

Any point within the feasible set, but below the feasible frontier will mean that not all of the 200 is distributed (eg 120 for Zoë and 40 for Yvonne would mean that some money is discarded)

Altruistic vs self-interested preferences

• Zoë will choose the point on the objective feasible set that gives her the highest subjective utility
• Her highest utility curve will be at a point of tangency with the feasible frontier
• Where this occurs will depend on whether she has altruistic preferences or has purely self-interested preferences
• If Zoë has altruistic preferences the point of tangency will be at point A where she keeps about 140 and gives Yvonne about 60
• If Zoë has purely self-interested preferences the point of tangency will be at point S where she keeps 200 and gives Yvonne 0
• In general, whether people behave altruistically may depend on the circumstances e.g. Zoë might be self-interested when she decides how to allocate her student budget, but altruistic when she wins the lottery.

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Applying self-interest and altruism to the prisoner’s dilemma

• We already know that if Anil is self-interested, his dominant strategy is T (the pesticide)
• We can show this in a different way using his indifference curves
• The left-hand panel shows Anil’s indifference curves when he is self-interested, and the four potential allocations in the game
• The monetary pay-offs for Anil correspond to the pay-off matrix on the right eg T,T gives him 2 and T,I gives him 4 (as Bala will also have a dominant strategy of choosing T, the dominant strategy equilibrium will be T,T)

Applying self-interest and altruism to the prisoner’s dilemma

• Suppose that Anil has altruistic preferences towards Bala, then his utility depends not only on his own monetary pay-off, but also on that of Bala
• First diagram: With these indifference curves, if Bala chooses T, Anil will choose I because (I, T) gives him more utility than (T, T).
• For altruistic Anil (I, T) is on a higher indifference curve than (T, T)
• Although Anil’s own monetary pay-off is lower at (I, T), he values the additional benefit to Bala
• Second diagram: If Bala chooses I, Anil will also choose I because (I, I) is on a higher indifference curve than (T, I).
• Although (I, I) gives Anil lower income, he prefers it because it doesn’t inflict damage on Bala.
• So I is altruistic Anil’s dominant strategy
• Altruism can make Anil’s behaviour cooperative
• Whether this happens depends on how altruistic he is. If he had cared a bit less about Bala’s income, his indifference curves would have been steeper and he might have made a different choice.

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Bala’s preferences are also relevant

• If Bala is similarly altruistic, he will choose IPC, too.
• Mutual cooperation will result in the good outcome, (I, I), being the dominant strategy equilibrium
• However, if Bala is still self-interested, he will choose T as before.
• Then the dominant strategy equilibrium will be (I, T), resulting in an allocation of 4 to Bala and 1 to Anil. Bala will benefit from high profits, while Anil (willingly) bears the monetary cost of Bala’s choice.
• Different equilibria are possible depending on the degree of altruism preference of each player
• If people care about one another, social dilemmas are easier to resolve and cooperative equilibria are possible in a prisoners’ dilemma

Repeated interactions

• Free-riding today on contributions by other members of one’s community may have consequences tomorrow or years from now.
• From year to year, some people may acquire a reputation for being uncooperative, which may influence the behaviour of others.
• Ongoing relationships are an important feature of social interactions that is not captured in the models we have used so far: life is not a one-shot game.
• Best responses may be different in a repeated game. Imagine how differently things would work out if the pest control game were repeated every season—even with self-interested preferences.
• Suppose that Bala had used IPC last season. What is Anil’s best response? He would reason like this: If I use IPC, like Bala did last season, then maybe Bala will continue to do so in future. If I use Toxic Tide—raising my profits this year—Bala is more likely to use Toxic Tide next year. So unless I am extremely impatient for income now, I’d better go for IPC.

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Impact of social norm of reciprocity and penalties

• When people engage in a common project—whether pest control, irrigation, or reducing carbon emissions—everyone has something to gain if they cooperate, but also something to lose when others free-ride.
• Even in large groups of people, repeated interactions, penalty mechanisms, social norms and social preferences (such as the norm or preference for reciprocity), can support higher levels of contribution to the public good.

Day care: Example of penalty crowding out social preferences

• It is common for parents to rush to pick up their children from day care. Sometimes parents are late, causing a negative effect on staff who have to work longer. 
• Two economists ran a field experiment introducing fines in some day care centres (the treatment group) but not others (the control group).
• Surprisingly, after the fine was introduced, the frequency of late pickups doubled.
• Possible Explanation:
• Before the introduction of fines, most parents were on time because they felt that it was the morally right to avoid inconveniencing the day care staff (altruistic concern for the staff)
• After the imposition of the fines lateness had a price and so could be purchased. If you paid the price, you had the right to be late, without consequence
• The use of a market-like incentive—the price of lateness—had provided what psychologists call a new ‘frame’ for the decision, changing it so that self-interest rather than concern for others was acceptable

When the fine was removed, parents continued to pick up their children late. They permanently adjusted their view of what was socially acceptable.

Cooperation in order to advance mutual benefits

• Cooperation⁠ means participating in a common project in such a way that mutual benefits occur. It need not be based on an agreement. Players acting independently may choose to behave cooperatively because of the following:
• They have social preferences: they are altruistic, or have a preference for fairness, or wish to reciprocate cooperative behaviour by others
• Their behaviour is guided by social norms: shared understandings that people in certain situations should cooperate
• They interact with each other repeatedly, allowing behaviour today to be rewarded, reciprocated, or punished in future
• In the one-shot prisoners’ dilemma, independent actions lead to an outcome that is not Pareto efficient. Then, the players may be able to achieve an outcome that everyone would prefer if they can reach an agreement. Anil and Bala might be able to agree that both would use integrated pest control, but they would need to find a way of ensuring that neither reneged on the agreement.

The Ultimatum Game – a sequential game

• Through the ultimatum game we can observe participants’ choices, and can investigate the participants’ preferences and motives, such as pure self-interest, altruism, inequality aversion, or reciprocity.
• How does the Ultimatum Game work?
• The Proposer is provisionally given $100.
• The Proposer decides how much money, y, to offer to the Responder; y can be anything from 0 to $100.
• The Responder can either accept or reject the offer.
• If the offer is rejected, both players get nothing.
• If the offer is accepted by the Responder, the Responder receives y and the Proposer gets 100 - y.
• In contrast to simultaneous games⁠ like the prisoners’ dilemma, in which players choose their actions at the same time, the ultimatum game is a sequential game⁠. One player, the Proposer, chooses an action first, followed by the Responder.

Ultimatum game

• The ultimatum game is used by economists to study social preferences
• It is a sequential game, where two players (the Proposer and the Responder) choose how to divide up a cash prize
• A sum of money ($100) is made available to a Proposer, but on condition that she shares it with the Responder

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• There are gains from cooperation for both if they can agree how to split the money
• Proposer has $100 to split.
• She makes a take-it-or-leave-it offer
• After observing the offer, the Responder accepts or rejects it
• Payoffs:

If the offer is rejected, both individuals get nothing.

If it is accepted, the split is implemented and both individuals get something

Testing for social preferences in the ultimatum game

• Rational self-interest: A responder who cares only about own payoffs would accept any positive offer, because something is better than nothing.
• Social preference: A responder who thinks that the proposer’s offer has violated a social norm of fairness, or that the offer is insultingly low for some other reason, might be willing to sacrifice the payoff to punish the Proposer.
• Experimental data:
• shows that very few people do behave with pure rational self-interest
• shows that people have social preferences and that they are prepared to pay a price to punish unfairness

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The Responder’s Reciprocity Motive

• Reciprocity a preference to be kind or to help others who are kind and helpful, and to withhold help and kindness from people who are not helpful or kind.
• Suppose $100 is to be split in an ultimatum game and there is a fairness norm of 50-50.
• If the Proposer proposes that the Responder get $50 or more, (y ≥ 50), the Responder will accept the proposal because the Proposer conforms to, or is even more generous than, the social norm.
• But if the offer is below $50, then the Responder may reject the offer to punish the Proposer for this breach.
• Rejection means that both receive nothing!
• So, if the Responder rejects the offer, it must mean that the satisfaction that she gets from punishing the Proposer must outweigh the satisfaction that she gets from accepting the Proposer’s offer.

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The Responder’s rejection equation

• Suppose the Responder’s willingness to punish the Proposer depends on the size of the breach of the 50-50 social norm.
• the Respondent’s satisfaction at rejecting an offer by the Proposer is shown by R(50 – y)
• e.g. if the offer y is 50 then R(50-50) = 0 willingness to punish the proposer
• e.g. if the offer y is 10 then R(50-10) = 40R willingness to punish the proposer
• which shows that
• (1) the smaller is y (the offer) the greater is the willingness to punish the proposer and
• (2) the greater is R (the social preference for reciprocity) the greater is the willingness to punish the proposer

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• Then, the Responder’s total level of satisfaction depends on two things:
1. The Respondent’s satisfaction at rejecting an offer R(50 – y), and
2. the gain from accepting the offer (y)

The decision to accept or reject an offer just depends on which of these two quantities is larger i.e. responder will reject an offer if y < R(50 − y) 

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To calculate minimum acceptable offer

• Responder will reject an offer if y < R(50 − y) i.e. if offer is less than satisfaction of rejecting the offer 
• To calculate her minimum acceptable offer we can rearrange this rejection equation like this:

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• R = 1 then Responder will reject any offer of less than $25 from the Proposer
• The cut off point of $ 25 is where the Responders two motivations (1) of monetary gain and (2) punishing the proposer for deviating from the social norm of fairness balance out,
• if Responder rejects the $25 she loses $25 but receives $25 of satisfaction from punishing the Proposer and making the Proposer’s payout = $0
• The more the Responder cares about reciprocity (as R rises), the higher the Proposer’s offers have to be e.g. if R = 2, then the minimum, offer must be 100/3 = R33,33 or it will be rejected by the Responder

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To calculate minimum acceptable offer

R is like the “weight” or “importance” that the Responder places on whether the Proposer makes a fair offer or not

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The more unfair the offer, the higher the Responder’s satisfaction will be when they reject the offer.

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If R is a large number, then the Responder cares a lot about whether the proposal is fair

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if R = 0 she does not care about the Proposer’s motives at all

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(50 – y) is the difference between what the Responder expects to receive and what is offered. Here, the 50 in R(50 – y) comes from (100 x 0.5 ) = 50

If the amount to be split were $400, and the expected split was 50-50, the expression would be R(400 x 0.5 – y ) = R(200 – y)

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Example of Ultimatum Game

• An experiment using the ultimatum game was run with a group of farmers in Kenya, and a group of students in the US. Proposers could offer 0, 10, 20, 30, 40, or 50% of the pie to Responders
• A purely self-interested Responder would accept any offer - that is always as good or better for themselves than rejection.
• So it is clear that many Responders were not motivated purely by self-interest.
• No one in either group accepted an offer of zero: they preferred to ensure that the Proposer got nothing as well.
• Kenyan farmers, in particular, were very unlikely to accept low offers, and almost half rejected offers of 30%. 

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From proposer’s perspective

• The Kenyan farmers made more generous offers
• 60% of them offered 40% or more. Only 11% of the students made such generous offers
• Their offers will depend on what they expect the Responders to do
• 89% of farmers accepted an offer of 40% of the pie, but lower offers were much less likely to be accepted (almost all offers of 20% or less can be expected to be rejected)
• If this is what the Proposer farmers can expect, then making a low offer is risky
• Even if they were not at all altruistic, they may have decided to offer 40% (to avoid rejection)

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Calculating estimated payoff

• The estimated payoff = the pay-off you receive if offer is accepted multiplied by the probability of acceptance
• For the Kenyan farmers offering 40% of the pie results in the maximum estimated payoff (0.96x60-=58%)
• Conclusions:
• The Kenyan farmers who make the Offers behave self-interestedly, given what they expected of Responders
• And the Responders’ behaviour was not consistent with pure self-interest as fairness and reciprocity matter to them

The Ultimatum Game with competition

• Consider an ultimatum game in which a Proposer offers a split of $100 to two respondents.
• If only one Responder accepts, that Responder and the Proposer get the split, and the other Responder gets nothing. 
• If no one accepts, the three players gets nothing.
• If both Responders accept, one is chosen at random to receive the split
• Results: The figure shows that individual Responders were less likely to reject offers when competing with another Responder
• The competition put the responders in a weaker position and the proposer in a stronger position – the responders therefore accept lower offer and the proposer can make lower offers with less fear of being rejected e.g. a 20% offer ill be rejected over 75% of the time if there is one responder but only about 25% of the time if there are two responders

When a Responder rejects a low offer this means they get a zero pay-off, but the Proposer may still get a positive pay-off if the other Responder accepts.

Rejecting no longer has the same impact on the Proposer. It is a less useful instrument for punishing a Proposer who is not following a social norm of fairness.

A Responder who cares about fairness cannot rely on the other Responder to reject low offers.

Summary of types of games discussed

In the next unit

• The role of institutions in social allocations

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• How to evaluate social outcomes: efficiency and fairness

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• How bargaining power affects the distribution of surplus

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