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EC403: Microeconomics 4
Assessed Essay
Katherine Anne Gilbert (201135988) Word Count: 1992 | 17th November 2014
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This essay focuses on the theme of social dilemmas where the rational behavior of an individual leads to a suboptimal outcome from the collective standpoint (Kollock, 1988). Robyn Dawes (1980) describes the theory as having two unique properties: 1) an individual will receive a higher payoff if they defect against the cooperative choice and 2) all individuals will receive a higher payoff if they all choose to cooperate than if they all choose to defect. This theme will be discussed through the example of Robert and Stuart where a model will be adopted to analyze their strategic situation through a simple yet insight-‐rich process. The game is played regarding the level of effort Robert and Stuart both choose to exert into a joint project. Game theory is defined as the interaction between people in society where interdependent behaviors cause the action of one person to affect another person’s well being, either positively or negatively (Watson, 2013). In this example, the effort level that Robert chooses to exert will have an impact on the value of the project which in turn affects the payoff to Stuart and vice versa. This interdependence shows a game is being played. When acting independently, individuals are assumed to act rationally with the incentive to maximize their own payout. Rational behavior has two unique properties: 1) from experience or knowledge, a belief is formed regarding the expected strategies of opponents 2) given this belief; a strategy is selected to maximize the expected payoff (Watson, 2013). When analyzing mixed strategies, the beliefs that are formed are a probability distribution over opponent’s possible effort level. As this example is a non-‐cooperative simultaneous-‐move game, players do not fully internalize their chosen value of effort and individuals are unsure what their opponents will choose therefore beliefs are central to individuals decision making process. The characteristics of this game show there is information asymmetry in the game, as players cannot observe opponents decision before taking theirs. This game will begin through a discrete example of Robert and Stuart selecting from a finite choice of either 1 or 2 units of effort to exert into the joint project. The project value is calculated by subtracting the individual cost function of Robert and Stuart from the value function, which measures the cost minus the benefit from exerting a particular unit of effort. The x value measures the effort expended by Robert and the y value measures the effort expended by Stuart.
Project value: V (x, y) = 5(x + y) + xy Project value depends on the level of effort exerted by Robert (x) and Stuart (y). As Robert’s cost function is 3x2 à Payoff function = 5(x + y) + xy – 3x2 As Stuart’s cost function is 4y2 à Payoff function = 5(x + y) + xy – 4y2
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When calculating the payoffs to each player from each chosen effort combination, the effort values are substituted into the individual’s cost function: For V (1, 1) ! 5(1 + 1) + (1 x 1) = 11 For V (2, 1) ! 5(2 + 1) + (2 x 1) = 17
For V (1, 2) ! 5(1 + 2) + (1 x 2) = 17 For V (2, 2) ! 5(2 + 2) + (2 x 2) = 24
Constructing the normal form for this game summarizes the players in the game, strategies available and payoff received to each individual based on the strategy combination the player’s choose (Gibbons, 1992). The columns correspond to the strategies of Stuart and the rows correspond to the strategies of Robert. Normal Form:
The underlining method identifies the rational best response made by individuals if their opponent chooses a particular effort level. By constructing the normal form, the Nash equilibrium can be identified: Robert chooses 1 unit of effort and Stuart chooses 1 unit of effort at V(1, 1). The Nash equilibrium represents a point in the game where players have no incentive to change their strategy and have mutually consistent best responses. For both players, strategy 1 is the dominant strategy therefore we can predict, due to rational behavior, that neither player will select strategy 2 (Gibbons, 1992). The identification of the best response in this game shows that both players have the incentive to underperform and reduce their individual effort to gain a higher payout. When analyzing the Nash equilibrium and normal form of this game, it is apparent that the rational behavior of Robert and Stuart does not necessarily imply coordinated behavior, as the Nash equilibrium is inefficient with both players able to gain a higher payoff by playing differently. The players realize they are jointly better off if they select 1 unit of effort, however individual incentives result in individuals defecting against this choice: Robert can gain a payoff of 13 instead of 7 and Stuart can gain a payoff of 14 instead of 8. In this game, individuals gain from being non-‐cooperative and have the incentive to free ride on the contributions of others.
R/S 1 2 1
8, 7
14, 1
2
5, 13
12, 8
V (1, 1) à Robert = 11 – 3 (1)2 = 8 Stuart = 11 – 4 (1)2 = 7
V (1, 2) à Robert = 17 – 3 (1)2 = 14 Stuart = 17 – 4 (2)2 = 1
V (2, 1) à Robert = 17 – 3 (2)2 = 5 Stuart = 17 – 4 (1)2 = 13
V (2, 2) à Robert = 24 – 3 (2)2 = 12 Stuart = 24 – 4 (2)2 = 8
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This result is similar to the Prisoner’s dilemma where individual incentives interfere with the interests of the group (Rapoport, 1965) resulting in group loss – this concept will be analysed later in the game. When opponents have complete freedom over their effort choices, the players no longer choose from a finite set of strategies and can undertake a continuous strategy. As the strategy spaces are continuous in this game, the game can be analysed by calculating best responses as opposed to through a payoff matrix. This is calculated by differentiating the individual’s payoff functions and re-‐arranging for x and y respectively. The derivative is set to 0 to find where the slope of this function is maximized at zero, which is the best response for each individual.
Substituting the best response function for y into the x function will find the payoff for Robert and substituting the best response function for x into the y function will find the payoff for Stuart.
The Nash equilibrium is at point (𝟒𝟓
𝟒𝟕, 𝟑𝟓𝟒𝟕), which represents the set of effort choices
for Robert and Stuart where both players are maximizing their payoff given the actions of the other player (Varian, 1987). Under this equilibrium, Robert will exert !"
!" units of effort and Stuart will exert !"
!"units of effort.
Robert: f(x) = 5(x + y) + xy – 3x2 f(x) = 5x + 5y + xy – 3x2 Differentiate with respect to x:
f’(x) = 5 + y – 6x Set f’(x) = 0 to find optimal point:
0 = 5 + y – 6x 6x = 5 + y x = 𝟓!𝒚
𝟔
BRr(x) = 𝟓!𝒚𝟔
Stuart: f(y) = 5(x + y) + xy – 4y2 f(y) = 5x + 5y + xy – 4y2 Differentiate with respect to y:
f’(y) = 5 + x – 8y Set f’(y) = 0 to find optimal point:
0 = 5 + x – 8y 8y = 5 + x y = 𝟓!𝒙
𝟖
BRs(y) = 𝟓!𝒙𝟖
Substitute y into equation x:
X = (5 + !!!!!)
6x = 5 + !!!!
48x = 40 + 5 + x 47x = 45 x = 𝟒𝟓
𝟒𝟕
Substitute x into equation y:
Y = !! 𝟒𝟓𝟒𝟕!
8y = 5 + 𝟒𝟓𝟒𝟕
376y = 235 + 45 376y = 280 y = !"#
!"#
y = 𝟑𝟓𝟒𝟕
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Substituting the x and y values into individual payoff functions and subtracting the effort costs, each partner receives the following payoff:
From the reaction functions calculated, the relationship can be diagrammatically presented by finding points on Robert and Stuart’s reaction function. This is calculated by setting the x and y values of Robert and Stuart’s reaction function to 0 as follows:
The following graph depicts the best-‐response functions of the two players. When calculating the equilibrium, the dominated strategies are removed and the lower and upper bounds converge to the point where the players’ best response functions cross, which is where the equilibrium point lies. These response functions are positively sloped as they are a complementary strategy set.
Robert: x = 𝟓!𝒚𝟔
To calculate reaction function: When y = 0, x = !!!
! = !
!
When x = 0, 0 = !!!! = -‐5
Points on reaction function: (5/6, 0) and (0, -‐5)
Stuart: y = 𝟓!𝒙𝟖
To calculate reaction function: When y = 0, x = !!!
! = -‐5
When x = 0, y= !!!! = !
!
Points on reaction function: (-‐5, 0) and (0, 5/8)
Payoff Robert: 5x + 5y + xy -‐ 3x2
5(𝟒𝟓𝟒𝟕) + 5(𝟑𝟓
𝟒𝟕) + (𝟒𝟓
𝟒𝟕)( 𝟑𝟓
𝟒𝟕) – 3(𝟒𝟓
𝟒𝟕)2
= 6.47
Payoff Stuart: 5x + 5y + xy – 4y2 5(𝟒𝟓
𝟒𝟕) + 5(𝟑𝟓
𝟒𝟕) + (𝟒𝟓
𝟒𝟕)( 𝟑𝟓
𝟒𝟕) – 4(𝟑𝟓
𝟒𝟕)2
= 7.00
(𝟒𝟓𝟒𝟕, 𝟑𝟓𝟒𝟕)
Robert’s Effort Level (x)
Stuart’s Effort Level (y)
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Game-‐theoretic analysis generally assumes that each player behaves rationally according to his preferences however; it does not take into account other motivations such as that of the altruistic player. Tony will be introduced to this example as the social planner who has the main objective of maximizing the total payoff of the project, which is then split between Robert and Stuart.
The function used to consider this game of social welfare is the project value function of both Robert and Stuart (5(x + y) + xy + 5(x + y) + xy) minus the two cost functions of Robert (3x2) and Stuart (4y2) to calculate the best responses to maximize the total payoffs from the game. ! V (x, y) = 5(x + y) + xy + 5(x + y) + xy – 3x2 – 4y2 ! V (x, y) = 10(x + y) + 2xy -‐ 3x2 – 4y2
The x and y effort values are calculated by substituting the x and y figures into one another as before.
Robert: f(x) = 10(x + y) + 2xy – 3x2 – 4y2 f(x) = 10x + 10y + 2xy – 3x2 – 4y2 Differentiate with respect to x:
f’(x) = 10 + 2y – 6x Set f’(x) = 0 to find optimal point:
0 = 10 + 2y – 6x 6x = 10 + 2y x = 𝟏𝟎!𝟐𝒚
𝟔
Stuart: f(y) = 10(x + y) + 2xy – 3x2 -‐ 4y2 f(y) = 10x + 10y + 2xy – 3x2 -‐ 4y2 Differentiate with respect to y:
f’(y) = 10 + 2x – 8y Set f’(y) = 0 to find optimal point:
0 = 10 + 2x – 8y 8y = 10 + 2x y = 𝟏𝟎!𝟐𝒙
𝟖
Substitute y into equation x:
X = !" ! !!"!!!!
!
6x = 10 + 2(!"!!!!)
48x = 80 + 20 + 4x 44x = 100 x = !""
!!
x = 𝟐𝟓𝟏𝟏
Substitute x into equation y:
Y = !" ! !!"!!!!
!
8y = 10 + 2(!"!!!!
) 48y = 60 + 20 + 4y 48y = 80 + 4y 44y = 80 y = !"
!!
y = 𝟐𝟎𝟏𝟏
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Payoff Robert: 5x + 5y + xy -‐ 3x2
5(𝟐𝟓𝟏𝟏) + 5(𝟐𝟎
𝟏𝟏) + (𝟐𝟓
𝟏𝟏)( 𝟐𝟎
𝟏𝟏) – 3(𝟐𝟓
𝟏𝟏)2
= 9.09 Payoff Stuart: 5x + 5y + xy – 4y2 5(𝟐𝟓
𝟏𝟏) + 5(𝟐𝟎
𝟏𝟏) + (𝟐𝟓
𝟏𝟏)( 𝟐𝟎
𝟏𝟏) – 4(𝟐𝟎
𝟏𝟏)2
= 11.36 Acting independently, Robert can gain a payoff of 6.47 and Stuart can gain a payoff of 7, however, when adding an altruistic player to maximize social welfare, Robert can gain a larger payoff of 9.09 and Stuart can gain a larger payoff of 11.36. This would suggest that Robert and Stuart should optimally act together to ensure they both gain larger payoffs than if they were acting independently. When comparing the effort levels of each player to the Nash equilibrium calculated when acting independently, the social planner states the players should exert more than double the effort that the players would exert when acting independently. Analysing in terms of the effort to payoff ratio, the individuals gain more from acting independently than if they were to exert effort at the social planners ideal value: Robert exerting 1 effort unit: 1.67 payoff acting independently and exerting 1 effort unit: 4 payoff acting socially optimally; Stuart exerting 1 effort unit: 9.5 payoff acting independently and exerting 1 effort unit: 6.3 payoff acting socially optimally.
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It is evident that the addition of the social planner to the game can result in effort levels chosen that will increase the total payoff function for the players. The choice for players is whether to cooperate with the social planners ideal outcome or whether to defect with the belief that their opponent will cooperate. As Kollock (1998) identified, the best outcome of a Prisoner’s Dilemma is unilateral defection of the first person, followed by mutual cooperation, mutual defection, and the worst outcome is the first person’s unilateral cooperation. The following analysis calculates the payoffs of Robert and Stuart if 1) Robert cooperates and Stuart defects and 2) if Stuart cooperates and Robert defects from the social ideal.
Normal Form:
The final normal form presents the payoffs to each individual based on the range of scenarios presented throughout this example. The Nash equilibrium is defection against the social planners ideal as it the dominant strategy for both players.
R/S C D C
11.4, 9.09
2.5, 14.7
D
13, 3.6
7.0, 6.5
When Robert cooperates with socially optimal input (𝟐𝟓𝟏𝟏) and Stuart defects:
Y = ! ! 𝟐𝟓𝟏𝟏!
Y = 0.91 Joint value of project: 5((𝟐𝟓
𝟏𝟏) + (0.91)) + (𝟐𝟓
𝟏𝟏)(0.91) = 18
Payouts:-‐ Robert: 18 – 3(𝟐𝟓𝟏𝟏)2 = 2.5
Stuart: 18 – 4(0.91)2 = 14.7
When Stuart cooperates with socially optimal input (𝟐𝟎𝟏𝟏) and Robert defects:
X = ! ! 𝟐𝟎𝟏𝟏!
X = 1.14 Joint value of project: 5((1.14 + (𝟐𝟎
𝟏𝟏)) + (1.14)( 𝟐𝟎
𝟏𝟏) = 16.9
Payouts: Robert: 16.9 – 3(1.14)2 = 13 Stuart: 16.9 – 4(!"
!!)2 = 3.6
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This example is similar to the prisoners’ dilemma. The dilemma is clear when noticing that mutual cooperation is superior and yields a better outcome than mutual defection of both players. This results in mutual defection not being a rational outcome because the choice to cooperate is more rational from a self-‐interested point of view. The value of the project suffers as partners only care about their own costs and benefits and they do not consider the benefit of their labor to the rest of the firm. The success of the enterprise requires cooperation and shared responsibility of the individuals. In this example, there are limits on external enforcement as the partners cannot write a contract and have no way of verifying the levels of effort exerted by each individual. Even if the players had the ability to communicate before the game and decide on a collective decision, there is no incentive for a rational player to follow through on the agreement as individuals could maximize by defecting. Players understand that increasing effort and therefore the value of the joint project will result in them only gaining a fraction of the joint benefit therefore the players are less willing to provide effort. The individuals therefore expend less effort than is best from the social point of view and because both players do so, they both end up worse off. As this example only considered a single play game, choosing to cooperate is not optimal for players. If the example incorporated repeated game analysis, including the punishment of the opponent for defecting would reduce the payoffs. This could result in players choosing to cooperate if costs > benefits by defecting.
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As individuals’ indifference curves depend on other people (Buchanan, 1962), this game results in the players imposing external effects on their opponent based on the decision they choose. Externalities are defined as an indirect consequence of an activity, which affect agents other than the originator without intension (Laffont, 2008). In this example, Robert and Stuart’s payoff depends on the effort exerted by the other player, which is an example of a positive externality or external benefit to the other player. As players increase their effort, the value of the overall project increases which increases the payoffs to both players. When calculating the effort to payoff ratio, there is a 2.8 payoff difference when Robert and Stuart exert one unit of effort: Robert gains 6.7 payoffs and Stuart gains 9.5 payoffs. It is clear that Stuart benefits more from exerting one unit of effort and therefore Robert exerts a higher positive externality on Stuart.
The existence of externalities can cause market failure if the price mechanism does not take into account the full social costs and benefits of production and consumption. In this game, there is a difference between the marginal private benefit and marginal social benefit resulting in the MSB curve lying above MPB. The effort exerted and therefore the market output is less than the socially optimal output as shown in Appendix 1 -‐ society could be better off and welfare increased by encouraging increased provision. There is therefore community pressure to make individuals pull their weight to bring the effort levels to the socially optimal ideal. A method of creating this pressure is to provide a subsidy to the individuals or reward the individuals to provide at this point [Appendix 2] bringing MSB = MPB.
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REFERENCES
Buchanan, J. & Stubblebine, W. (1962). Externality. Economica. 29 (116), 371-‐384. Dawes, R. (1980). Social Dilemmas. Annual Review of Psychology. 31, 169-‐193. Gibbons, R (1992). A Primer in Game Theory. London: Pearson Education Limited. Kollock, P. (1998). Social Dilemmas: The Anatomy of Cooperation. Annual Review of Sociology. 24 (1), 183-‐214. Laffont, J. (2008). Externalities. The New Palgrave Dictionary of Economics. 2. Rapoport, A (1965). Prisoner's Dilemma: A Study in Conflict and Cooperation. USA: The University of Michigan Press. Varian, H.; 1987; “Intermediate Microeconomics – A Modern Approach; Norton, Chapter 27 pp. 508 Vicarick. (2011). Externalities Graphs: How I Understand Them. Available: http://www.slideshare.net/vicarick/externalities-‐graphs-‐how-‐i-‐understand-‐them. Last accessed 14th Nov 2014. Watson, J (2013). Strategy: An Introduction to Game Theory. 3rd ed. New York: W. W. NORTON & COMPANY. ADDITIONAL BIBLIOGRAPHY Carmichael, F (2005). A Guide to Game Theory. ENGLAND: Pearson Education Limited. Dixit, A., Skeath, S. & Reiley, D (2009). Games of Strategy. 3rd ed. New York: W. W. Norton & Co. Osborne, M (2004). An introduction to game theory. New York: Oxford University Press.
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APPENDICES Appendix 1
(Vicarick, 2011)
Appendix 2
(Vicarick, 2011)