Political science major

Hume’s Marsh – Draining game • No matter what Farmer B does, Farmer A always gets a higher payoff if he chooses not to drain. The reasoning is precisely the same • No matter what Farmer A does, Farmer B always gets a higher payoff if he chooses not to drain. Stag Hunt game (Rousseau) • An alternative vision of the problem of social cooperation is provided by the Stag Hunt Game Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • What is the most preferred outcome ? Is there another outcome in which neither player has an incentive to alter his strategy ? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • The most -preferred outcome for both players is (Stag, Stag), for which each player receives a payoff of 3. This outcome is an equilibrium inasmuch as neither player wishes to alter his strategy when he believes the other player will be playing Stag. • (Hare, Hare) is also a stable outcome or equilibrium because if A believes that B is going to play Hare, than A’s best response is also to play Hare. Likewise, if B believes A is going to play Hare, than B will play Hare, too. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Does either player end up doing better playing either Stag or Hare no matter what his partner chooses to do ( as in the marsh -draining game)? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Unlike the marsh -draining game,neither Stag nor Hare is always the optimal strategy regardless of the strategy employed by the other player. If a player believes his partner will play Stag then his best option is to play Stag. But if a player believes his parter will play Hare, than his best response is to play Hare. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • How certain must A be that B will playing Stag to do the same ? Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • Let’s define pB as the probability that B plays Stag . Then A’s expected utilities associated with the two strategies are: • EUA[Stag] = pB ⋅ 3 + (1 − pB ) ⋅ 0 = 3pB • EUA[Hare] = pB ⋅ 1 + (1 − pB ) ⋅ 1 = 1 Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Stag Hunt game (Rousseau) • A will wish to play Stag when EUA[Stag] > EUA[Hare] , namely 3pB > 1 or pB > 1/3 Achieving the most -preferred outcome in this game then requires that both players believe that the other player will play Stag with at least probability 1/3. One interpretation of this is that the equilibrium depends on each player’s conjecture about the other’s behavior . Another interpretation is that the players must trust one another to play a certain outcome (at least up to a point) in order to secure the socially -optimal outcome. Hunter B Hunter A Stag Hare Stag 3,3 0,1 Hare 1,0 1,1 Chicken game ( Rebel Without a Cause, 1955) • Another famous coordination Game. Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • What is (are) the most preferred outcome (s) ? What are the «pure» equilibria outcomes ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game ( Rebel Without a Cause, 1955) • Does either player end up doing better playing either Go Straight or Swerve no matter what his competitor chooses to do ( as in the marsh -draining game)? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) • How certain must A be that B will playing Swerve to play the Go Straight ? Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • Let’s define pB as the probability that B plays Go Straight . Then A’s expected utilities associated with the two strategies are: • EUA[Go straight] = pB ⋅ -5 + (1 − pB ) ⋅ 3 = 3 -8pB • EUA[Swerve] = pB ⋅ -1 + (1 − pB ) ⋅ 0 = -pB Chicken game (« Rebel Without a Cause», 1955) Racer B Racer A Go straight Swerve Go straight -5, -5 3, -1 Swerve -1,3 0,0 • A will wish to play Go Straight when • EUA[Go Straight] > EUA[Swerve] , namely • 3 -8pB > -pB or 3 >7pB or pB <3/7 Gangs’rewarding cooperation • In order to prevent the prisoner’s dilemma outcome , criminal organizations can also reward the “cooperation” ( do not confess) for instance by looking after an individual’s family while the criminal is in prison. Gangs’rewarding cooperation • Suppose that a bonus of is given to a criminal who cooperates but whose partner defects, while a payoff of is given to a criminal who cooperates and whose partner also cooperates. a) Rewrite the payoff matrix b) For what values of and is cooperation an equilibrium ? c) For what values is it the only equilibrium ? Gangs’rewarding cooperation • Mutual cooperation is an equilibrium if ≥ 1. For what values is it the only equilibrium ? • Mutual cooperation is the only equilibrium if ≥1 Apartment cleaning • 4 friends (X, Y , Z ,W) live together in a college apartment and must work together to clean common areas . Outcome is dichotomous and has the feature of the following collective action problem . Assume that B (utility coming from cleaniless )>C ( cost of cleaning ) Apartment cleaning • What are the possible equilibrium outcomes when all 4 friends must contribute ? Which do you think is likely ? Apartment cleaning • What are the possible equilibrium outcomes when all 4 friends must contribute ? Which do you think is likely ? • There are two outcomes which are equilibria: 1) everyone contributes or 2) no one contributes. If everyone contributes, each individual secures a benefit B and pays cost C. Thus, their net payoff is B -C > 0. 1) With everyone contributing, if one person decides to not contribute, than the apartment is not cleaned. Those contributing then get net payoff -C, while the person who didn’t contribute gets a payoff of 0. Because B -C > 0, the now non -contributor is worse off than she had been when she contributed along with all of her apartment mates . 2) If no one is contributing, than each player earns a payoff of 0 . No player will wish to unilaterally start contributing because that will only lead to them paying the cost of contribution without securing any benefit, hence the net payoff goes from 0 to -C, and 0 is preferable. Apartment cleaning • What are the possible equilibrium outcomes when only 2 of the 4 friends must contribute to clean the apartment (k=2)? Can you predict which outcome will occur without further information ? Apartment cleaning • What are the possible equilibrium outcomes when only 2 of the 4 friends must contribute to clean the apartment (k=2)? Can you predict which outcome will occur without further information ? • If only two members are required to clean the apartment, than any combination of the two apartment mates contributing is an equilibrium. XY; XZ; XW; YZ; YW; WZ • Imagine for example, suppose X and Y contribute and Z and W don’t. Z and W certainly don’t want to start contributing because they are already getting B without having to pay C. Nevertheless X and Y still prefer B -C to 0, which is what will occur if either one of them decides to not contribute. There are 6 possible ‘cooperative ’ equilibria. • But there is also one ‘non -cooperative’ equilibrium as no one wants to be a sucker and start contributing on their own, because one person cleaning is insufficient to fully clean the apartment. Thus, there are 7 possible equilibria, and it is hard to predict beforehand which will occur . • However , compared to the previous condition, now each roommate has ½ probability to enjoy the cleaniless without any effort Apartment cleaning • How might the prediction change ( when k=2) if B increases or C increases ? What about if B is different for different members of the group ? Apartment cleaning 1)How might the prediction change ( when k=2) if B increases or C increases ? 2) What about if B is different for different members of the group ? 1) Higher B or lower C means that the incentives to coordinate on a cooperative equilibrium are greater . 2) we might expect to see high -B individuals exploited by low -B members, who free ride confident in the knowledge that the very high benefits secured by high -B individuals will motivate them to coordinate on cleaning the apartment. Typology of goods Excludability Yes No Non Rivalrous Yes No Cable or satellite TV «Premium» version on line Journal A Pizza The Global Positioning System (GPS) Public beaches Knowledge Street lighting National Parks One individual public good provision game with mixed strategies • Suppose that there are n individuals who desire a collective good that yields benefit B to all n individuals . Provision of the good requires only one individual (k=1) to expend C to provide it (B>C). Show that there are n possible sets of «pure» strategy equilibria ( each player i plays « contribute » or « Don’t contribute » with probability 1) One individual public good provision game with mixed strategies • The n possible pure strategy are all the same: one individual contributes and no other player does. • If one individual is contributing, no one else wishes to add on their own contribution because the group benefit B has already been secured by all, and adding a contribution would only waste C units of utility. • The net payoff for the sole contributor is B -C; however, we know this is a positive quantity which the contributor prefers to receiving no B and paying no C ( that would be equal to 0) , which is what will occur if he withdraws his contribution. • There is one ‘single -contribution’ equilibrium for each of our n individuals, and hence n pure strategy equilibria. One individual public good provision game with mixed strategies • Now suppose that all players are playing an indentical mixed strategy . In other terms they probabilistically choose whether to play C (oop .)or D( on’t ) . Call p the probability that any one player plays C. a) Show that for any player i, if he does not contribute , the probability that the good is supplied by some one else is 1 -(1 -p) n -1 One individual public good provision game with mixed strategies if A represents some event occurring and A’ represents that event not occuring , then Pr (A’) = 1 -Pr(A) 2. If A and B are two independent events, then Pr (A and B both occur) = Pr (A)* Pr (B). The probability that a player contributes is p, the probability of that player not contributing is (1 − p).

Each player makes their decision independently , so the probability that every player but i does not contribute is (1 − p)(1 − p)…(1 − p) = (1 -p) n -1 If (1 -p) n -1 is the probability that everyone but i does not contribute, then 1 − (1 − p) n−1 is the probability that at least one person (excluding i for the moment) contributes. One individual public good provision game with mixed strategies • Equate the expected utility for i of contributing with the expected utility of not contributing . • Solve the expression you found for p • Show that p is decreasing in C, increasing in B and decreasing in n. One individual public good provision game with mixed strategies • Equate the expected utility for i of contributting with the expected utility of not contributing . • The utility for i of contributing is : EUi [Contribute] = B − C; • while the expected utility for i of not contributing is EUi [Don’t] = B[1 − (1 − p) n−1 ] One individual public good provision game with mixed strategies • When these two expected utilities are equal, i is ambivalent about whether to play Contribute or Don’t, which opens up the possibility of playing a probabilistic mixture of Contribute or Don’t, otherwise known as a mixed strategy. • A player is only willing to play this probabilistic mixture of strategies when the payoffs associated with each strategy are exactly equal. One individual public good provision game with mixed strategies • B − C = B[1 − (1 − p) n−1 ] • B(1 − p) n−1 = C • p = 1 − (C/B ) 1/n− 1 • B > C, therefore C/B < 1. As C increases, the second term gets larger so p gets smaller. • An intuitive interpretation of this is that as the costs of contribution increase, each person is less willing to contribute, holding all other factors constant. • Similarly, as the benefits of contribution (B) increase, individuals are more likely to contribute, reflecting the extra gains from contribution. Finally, as the number of individuals increases, each person is less likely to contribute. • This makes sense, because as more individuals contribute with some probability p the more likely it is the good will be provided (only one needs to contribute, after all so each person can relax a little bit. Externalities • A factory located in a small village produces a good with increasing marginal costs MC(q) = 12 + q; so the first unit costs 13 , the second 14 etc. This firm can produce at most 15 and no fractional amount can be produced . The market price for the good is p=20$ and firm’s level of production does not affect this price . Assume that the factory owner maximizes profit and her utility is measured in dollar . Profit is calculated by summing up differences between the price and the marginal cost of each unit produced . • The factory is noisy and interfere with the practice of a neighboring doctor . For every extra unit produced the doctor loses $2 worth of profits . • The doctor’s welfare depends only on his profits , which are 50 -2q Externalities 1) How many units of the good will the factory produce if it ignores the externality imposed on the doctor in its profit maximization ? What will be the aggregate social utility ( factory’s and doctor’s total profits ) ? Externalities (1) • If the factory ignores the external effects of its production, then it will produce up to the point where the marginal revenue of an extra unit equals the marginal cost of an extra unit. The marginal revenue for each unit is $20, and is invariant to the level of production. The marginal cost increases steadily, and will equal $20 when q = 8, which yields a profit of 7+6+5+4+3+2+1+0 = $28. • At this level of production, the doctor’s profits are 50 − 2q = $34. Therefore, aggregate social utility is 28 + 34 = $62 unit revenue cost profit 1 20 13 7 2 40 27 13 3 60 42 18 4 80 58 22 5 100 75 25 6 120 93 27 7 140 112 28 8 160 132 28 9 180 153 27 10 200 175 25 11 220 198 22 0 5 10 15 20 25 30 0 2 4 6 8 10 12 Titolo del grafico marginal revenue marginal cost profit Externalities 2) Identify the level of production that is socially most preferred , in other terms that maximizes aggregate social utility. Externalities (2) • Social utility, S, has been defined as the sum of the factory’s and doctor’s profits. • S(q) = 20 − σ = 1 12 + + 50 − 2 . • This is maximized where q = 6. • We need to consider the marginal profit for an extra unit of production for the factory owner against the marginal cost of that extra unit imposed on the doctor. For example, if the factory increases q from 0 to 1, this garners the factory $7 units of profit while imposing a cost of only $2 on the doctor. • When q = 6, the extra two dollars of profit for the factory are exactly cancelled out in the social utility function by the two dollars of loss in the doctor’s profits. • At this level of production, S = 120 -93+38 = $65 -10 0 10 20 30 40 50 60 70 0 2 4 6 8 10 12 Titolo del grafico Marginal profit Marginal d’s cost Social utilityunit revenue marginal revenue marginal cost cost profit Doctor’s tot utility Marginal profit Marginal d’s cost Social utility 1 20 20 13 13 7 48 7 2 55 2 40 20 14 27 13 46 6 2 59 3 60 20 15 42 18 44 5 2 62 4 80 20 16 58 22 42 4 2 64 5 100 20 17 75 25 40 3 2 65 6 120 20 18 93 27 38 2 2 65 7 140 20 19 112 28 36 1 2 64 8 160 20 20 132 28 34 0 2 62 9 180 20 21 153 27 32 -1 2 59 10 200 20 22 175 25 30 -2 2 55 11 220 20 23 198 22 28 -3 2 50 Externalities 2) Propose a government taxation scheme that will lead to socially preferred outcome … Obviously a tax of $2 per unit of production levied against the factory would lead to the socially optimal amount of production. The factory would only produce up to 6 units (after this point, the marginal profit of an extra unit turns negative) Donation of time • 5 civic -minded patrons of a public library contemplate donations of time to its annual fundraiser . Each individual i bases his or her decision of how much time to donate, x i, on the following utility function : Where namely the total amount of time given by all library patrons and is the cost of losing x i of one’s leisure time Donation of time • What is the socially optimal amount of time donated ? Determine this by summing the utility functions of five individuals , and finding the q that maximizes this function . Donation of time • What is the socially optimal amount of time donated ? Determine this by summing the utility functions of five individuals , and finding the q that maximizes this function . • First we need an expression for society’s utility, (S ). We can find this by adding up the utilities of the individuals who we will index by j 2 (1; 2; 3; 4; 5): Donation of time • We can maximize with respect to q to find the socially optimal net contribution (let’s call it q*). Because all of our individuals are identical, we can then divide this by 5 to find the socially optimal individual contribution x* i At the maximum of this function, this derivative will equal zero. We can solve this for q to find that The socially optimal individual contribution is thus Donation of time • Imagine that an individual assumes that everyone else will contribute no time. How much time will this individual donate ? Is it at the socially optimal level ? Donation of time • Imagine that an individual assumes that everyone else will contribute no time. How much time will this individual donate ? Is it at the socially optimal level ? • If some individual i believes that no one else will contribute, then he will behave as if ; his expected utility will be ;Maximizing this with respect to x i, ; x i=1 < 1.710 , the socially optimal individual contribution < 8.550, the “social” optimal level. Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? If i assumes that the other individuals will collectively supply .8 units he will believe that ; Maximizing with respect to x i, we get this expression which equals zero where EUi is maximized ; x i=.2 Donation of time • Imagine that each individual assumes that the other 4 members will donate 0.8 units of time. Does this individual donate more or less time than before ? Why this ? The level of contribution .2 turns out to be a symmetric equilibrium , where each individual gives .2 towards the cause. This results in only 1 unit total of time donated, which is far less than the socially optimal level of 8.550 . Each individual gives only .2 , rather than the 1.710 units which would maximize the group’s welfare. The individual’s belief that the others will collectively provide .8 units is confirmed in equilibrium. Thus , this belief is “rational”.

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