Evolutionary Game Theory and Behavioral Economics

by Shirley Jean-Baptiste, Nancy Phean, Thanh To

Introduction

With the increasing importance of mathematics in economics approaches, our project is to focus on the application of quantitative science to behavioral economics. Specifically, we will investigate the more sophisticated tools needed to develop more complicated economic models as the economic thoughts evolve. The economic paradigms of interest are the Walrasian paradigm, which dominated in the third quarter of the 20th century, and another economic paradigm which we refer to as “Bowles’ paradigm” for the sake of easy distinction (it does not necessarily mean that Samuel Bowles invented this paradigm). Specifically, we will explore why evolutionary game theory is more appropriate to be applied to economics than classical game theory. The three following aspects would be our main concern:

• The domain: exogenous versus endogenous preferences and institutions

• Contractual versus non-contractual assumption

• Equilibrium selection problem

Background

Walrasian Paradigm

This paradigm assumes that

• individuals choose actions based on the far-sighted evaluation of their consequences based on preferences that are self-regarding and exogenously determined

• social interactions take the exclusive form of contractual exchanges

• increasing returns to scale can be ignored in most applications and institutions do not evolve

• represents economic behavior as the solution to a constrained optimization problem faced by a fully informed individual in a virtually institution-free environment

Bowles' Paradigm

This paradigm assumes non-contractual social interactions, adaptive and other-regarding behaviors, and generalized increasing returns.

Game Theory

Game Theory is a way of modeling of strategic interactions, which means the consequences of each individual's actions are dependent on the actions taken by others and it requires detailed attention to the institutional environment in which the game takes place. Economists use game theory in order to speculate outcomes in regards to individual behaviors.

Classical Game Theory

The Walrasian paradigm approaches a classical game theory. It is also known as the conventional game theory. Classical game theory gives emphasis to the idea that the players in the game are forward-looking. There are two main solution concepts used in the classical game theory, which are: Dominance and Nash Equilibrium. These two solutions are based on a best response strategy.

• Dominance uses predictions of outcomes in games such as prisoner’s dilemma, in which every player acts according to what other players would do; it is a process of elimination.

• Nash equilibrium’s solution concept to a game assumes that each player knows the equilibrium of the other player. Using this knowledge, each player has no incentive or benefit to change their choice. Therefore, the payoffs of their choice are known as the Nash equilibrium. In addition, there are no endogenous source of change because each player are not interested in altering their decision.

Bowles’ critique: Classical game theory assumes that this is the standard way the game will be played. If a player was playing through reasonable play, there is little reason to think that this would lead to Nash equilibrium. Furthermore, not every game has a nash equilibrium and some games have multiple nash equilibria, hence outcomes cannot necessarily be predicted.

Classical game theory has addressed this problem of multiple nash equilibria by placing restrictions on player's behavior based on rationality attempting to narrow the possible outcomes. These restrictions were called refinements, which include noncredible threats. These noncredible threats were generally outcomes that were not the best response, or outcomes that resulted without the use of all available information.

Evolutionary Game Theory

Samuel Bowles supports the evolutionary game theory model. The evolutionary game theory typically assumes that individuals have limited information about the consequences of their actions. They update their beliefs by trial-and-error methods using local knowledge based history. In other words, they use past experiences that they have experienced themselves or that they observed from others in order to update their beliefs. In contrast to classical game theory, players in the games are “intellectually challenged” and backward looking.

In addition, this game theory takes into account the complex human interaction because humans act strategically. There are also many more important interactions: labor markets, credit markets, markets for information and goods of variable quality. Evolutionary game theory also approaches: modeling of chance, differential replication, out of equilibrium dynamics, and population structure.

Another reason to reject classical game theory is that it assumes equilibrium can be settled solely by game theory itself and ignores history of the players, however social outcomes are often influenced by recent past. Society cannot be well-modeled by a single game, especially one with an unchanging structure. Overall, game theory has a narrow scope of institutions. In reality, games overlap and intersect because people participate in a variety of social interaction. This is important because players learn through the structure of the game, which results in cultural evolution. This then affects how the players play the game and how they play other games that they are involved in. People also update their behaviors according to change and local information.

Moreover, some characteristics of Evolutionary modeling includes: populations are structured hierarchically and differential replication can take place at more than one level; process of differential replication is typically taking place at many levels simultaneously: within individuals, among individuals and among groups.

The Hawk Dove Game is commonly applied to culturally or genetically transmitted human behavioral traits such as aggression and sharing.We know that hawks are fighters and doves are peace makers, so when doves meet they share a prize, whereas when hawks meet they fight over the prize, inflicting costs on one another, and when a Hawk meets a Dove, the Hawk takes the prize inflicting costs on doves. Thus, both doves and hawks are better off with fewer hawks. However, the Nash equilibrium of p* = v/c, in which v is the prize to be divided and c is the cost of losing a fight is a weak best response and is not a desirable outcome. Based on Hawk Dove Game, the average payoff for both hawks and doves is when p=0 when there are no hawks at all. In other words, hawks and doves are better off with fewer hawks. Overall, if there is only one strategy to replicate, it should remained unchanged, however the population may be invaded by mutants, which the conventional model does not take into account. In other words, for a strategy to be evolutionarily stable, it must have the property that if almost every member of the population follows it, no mutant with a different strategy can successfully invade. There can be a society with a mixture of both hawks and doves and a mutant.

This game is typically used to explore the issues that arise in regards to ownership and division of resources. John Maynard Smith (1974), who is a biologist proposed the solution to the Hawk Dove Game, which was to find a way to reduce the number of interactions between hawks and doves that results in conflict. He suggested that “if owner plays Hawk, if intruder play Dove” labeled “Bourgeois”, where everyone's territory is made prevalent. In other words, proprietorship is never in question and everyone has an equal probability of being an owner.

Example: When Bourgeois meets a Hawk, half of the time it is not the owner and it then acts like a Dove, thus avoiding a fight. The other half of the time, the Bourgeois is an owner and they fight, but with the probability of winning as 1/2, the expected pay off would be (v-c)/4.

Overall, Bowels points out the the key success of a Bourgeois is the use of additional information to create asymmetry among players.

Problems that arise from Smith's solution:

• Ambiguity of possession arise

• Private property rights could have evolved spontaneously

• Institutional crowding out

• Invisible hand arguments are misleading when applied to institutions and behavioral traits

A similar analogy to this game theory involves a "chicken" in which drivers never swerve, so when they meet, they crash. But, when the drivers meet the chicken, the chicken swerves and they both benefit.

Walrasian's Paradigm versus Bowles' Paradigm

The Domain: Exogenous Versus Endogenous Preferences and Institutions

Bowles argues that many behaviors are best explained by social preferences: “in choosing to act, individuals commonly take account not only of the consequences of their actions for themselves but for others as well” (p.96). In other words, people not only care about the consequences of their action, but also about the intentions of others. An important aspect of social preferences is reciprocity, which is essentially people treat others according to the way the person behaves towards them or others. For example, people are generous to those behave well and people punish those who behave poorly. People act according to this reciprocity motives despite the fact that there are no incentives and may inflict costs on themselves. Other social preferences considered by Bowles includes: inequality aversion, envy and altruism.

In contrast, F.Y. Edgeworth, who is a founder of the neoclassical paradigm assumes self-interest and self-regarding preferences defined over outcomes. In the conventional approach, it is generally assumed that people act out of self-interest, while viewing that the other person is also acting out of self-interest.

In addition, individuals act according to social rules and norms with our limited cognitive resource. This contrast with conventional view “in which behavior is the result of often quite demanding individual cognitive processes addressing both evaluative and causal issues. This conventional individual cognition-centered view excludes behavior based on visceral reactions such as fear, disgust and much more, habit or evolved rules of thumb." It also presumes that people are able to predict what others will do and about the way the world works.

Bowles emphasizes that preferences are situationally specific and endogenous: "because endogenous preferences involve learning or genetic changes, behavior in the same situation changes over time” (p.97). Moreover, people evaluate outcomes through certain perspectives in regards to the current state or the current state experienced by a member of the group. Next, social institutions influence social interactions through the people we meet and the things we do, thus motivations are guided by culture.

On the other hand, conventional approaches generally view preferences as unchanging and that individuals seek to maximize their utility, which Bowles think limits economists view of both behavior and social evaluation. Conventional approaches seems to ignore other behavior aspects such as the fact that things change and the influence of the situation on behavior.

One of the conventional approach models individual behaviors through utility function:

• U = U(x,y,z), which describes a list of of goods consumed by individuals.

The utility function is organized by the relationship between preference and indifference. The ordering of this function is also transitive and time invariant. For example, when prices change exogenously, individuals act accordingly, which is a form of beng rational. Hence, those who act outside of the order are considered irrational. The conventional model does take into account risks and uncertainty in which individuals are assumed to maximize their expected utility.

Although the conventional model of utility function take into account behavior, it does not match recent empirical research about behavior. Bowels points out that the indifferent aspect in the conventional approach ignores the idea that changes in taste occurs. For example, individuals may not like one thing at first, but they may grow to like it or become accustomed to it. In other words, preferences may adjust to certain situations. According to the discounted utility model, you would continue to be indifferent towards that good, which is also called stationarity property:

• U(y+x) - U(y) = {(U(y+x') - U(y)d^n

Daniel Kahneman, Amos Tversky, Richard Thaler and their co-authors suggested prospect theory that takes into account four aspects that the conventional paradigm does not which are: people don’t evaluate risky decisions based on expected utility, framing (situations often frame choices), experienced hedonic utility, and framework that behaviors are situation-dependence. If the utility function was to explain actual behavior, it should argue that changes in states or events rather than states.

Bowles suggest that social preferences are these other regarding and processing regarding reasons for behavior (e.g. people hate being taken advantage of, people care about others as opposed to maximizing utility).

Process-regarding preference: individuals' evaluation of the state depends on the condition on how it came about rather than the intrinsic characteristics of the state. (e.g. the desire to help the homeless/poor as a result of bad luck not because they were lazy)

Other-regarding preference: individuals' evaluation of the state depends on how it is experienced by others which may include: spite, altruism, and caring about the relationship among the outcomes for oneself and others. (e.g. the desire to punish those who has done harm to you)

Contractual versus Non-contractual Assumption

Edgeworth Box

One of the main assumptions in neoclassical economics is that agents pursue self-interests. Thus, exchanges between agents must occur on the contract curve that represents equilibrium outcomes, which can be geometrically illustrated with the Edgeworth box.

Let us consider the following example. In Figure 1, we have

• Two agents: John and Karl.

• Two commodities: X and Y.

• The width of the box represents the fixed social amount of X available, denoted as α.

• The height of the box represents the fixed social amount of Y available, denoted as β.

• The fixed dimension of the Edgeworth box means that we assume no production takes place here.

• John’s origin O_j is at the lower-left corner of the box. Karl’s origin O_k is at the upper-right corner of the box.
• The axes represent John and Karl’s consumption quantities x_j, y_j and x_k, y_k, respectively. The amount of goods consumed by each agent is measured by the distance from each person’s corresponding origin in the direction of the axes (as indicated by the arrowhead of the axes). Thus, we have α = x_j + x_k and β = y_j + y_k, which means if John increases his consumption of goods X then Karl’s consumption of goods Y must decrease, unit by unit.

• John’s indifference curves are denoted as U_j’, U_j’’, U_j’’’, …, where as Karl’s indifference curves are denoted as U_k’, U_k’’, U_k’’’, … Recall that indifference curves of each agent are parallel curves that represent their equally preferable consumption bundles. As we move northeast John’s utility will increase. In contrast, Karl’s utility increases if we move southwest.

• The shaded region is called Region of Mutual Advantage, which contains the allocations that are more preferable than the endowment position E by both parties. The endowment position shows the initial allocation of the two goods between John and Karl.

Every point in this box represents an allocation of X and Y between John and Karl. Trading occurs when there is a shift from one allocation point to another point. Given that the agents are rational, voluntary trading means they would scarify an amount of goods X to consume more of goods Y, and vice versa, in the condition that each person’s utility will increase. Next, we will introduce the price of X and Y to the model (as illustrated in Figure 2).

Karl: from E to Q_k --> increase Y, reduce X

John: from E to Q_j --> increase X, reduce Y

Note that our two parties are price-takers and have no influence on the prices. Assume we have P_x and P_y that give us the line segment KL as shown in Figure 2. Recall that the slope of KL has an absolute value of P_x/P_y. Hence, a different price ratio will give a different budget line. The line segment KL serves as the budget line for both John and Karl. It follows that KL also indicates the upper limit of both John and Karl’s market opportunities. Obviously, Q_k is the consumptive optimum for Karl whereas Q_j is the consumptive optimum for John. However, Q_k and Q_j are two separate points. Given that we have a fixed total amount of goods X and Y, it is not possible that both John and Karl will obtain their consumptive optima. Trading cannot take place because of the low price ratio in this case, i.e. X is too cheap relative to Y (the amount of X John wants to take is greater than the amount of X Karl is willing to offer). This leads to the notion of an equilibrium price ratio, which is the price ratio in which Q_k and Q_j will coincide with each other. In Figure 2, this price ratio is illustrated as the line segment K’L’. At this price ratio, the amount of commodity X Karl willing to offer equals the amount of X John willing to purchase.

“In the Edgeworth box, the competitive equilibrium allocation of the two commodities (1) lies in the region of mutual advantage, and (2) represents a mutual tangency of both traders’ indifference curves with one another and with a common budget line.” (Hirshleifer, 1980)

Mathematically, consumptive optimality is obtained when the following conditions are satisfied:

MRSC_j = P_x/P_y = MRSC_k (1)

x_j + x_k = α (2)

y_j + y_k, = β (3)

Constraint (1) ensures the tangency between the budget line and the difference curves of John and Karl. Constraints (2) and (3) ensure that the consumptive optima for John and Karl will coincide.

"The Walrasian general equilibrium theory was one of the most elaborate expositions of the neoclassical paradigm. The theory hypothesized that the principle of marginality could be used to prove that the price mechanism, under the standard neoclassical set of assumed ideal conditions, was capable of clearing all markets and, consequently, lead to the emergence of a general market equilibrium.

The goal of all economic activity is to provide utility for its participants. The Walrasian general equilibrium theory was mainly concerned with the question of prices as a clearing mechanism, and not so much with the utility aspect. This was in line with the general trend of the neoclassical period, which saw the questions of how welfare and wider social goals connected to the economic processes pushed to the background." (Sandbec)  

The Model of Labor Market

Definition/background: The supply of available workers in relation to available work, they function through the interaction of workers and employers; from the neoclassical perspective, the forces of supply and demand determine the wage rate (price) and the number of people employed (quantity). Supply and demand set the price and quantity and it has a persistent level of unemployment according to Bowles

Work effort cannot be contracted because an employee’s effort is not verifiable because the employer is not always familiar with it. This is known as the employment relationship. As a result of this relationship, models based on incomplete contracting for effort or other aspects of the labor exchange help explain how a competitive equilibrium could exhibit involuntary unemployment and during this process the standard theories of the labor market and the firm have been transformed.

The importance of reciprocity motives and other social preferences in explaining why firms do not sell jobs underlines the futility of simply introducing incomplete contracting into an otherwise unaltered Walrasian framework

If the difficulty of monitoring labor effort differs across technologies, the choice of technology will be influenced by the nature of the labor discipline problem; technologies and institutions co evolve and influence the development of the other. Such aspects of the labor discipline environment as prevailing norms and whether terminated workers have access to unemployment insurance, and other influences on the worker’s effort choice will affect the profitability of alternative technologies. This view goes against the standard model where the choice of technology responds to factor scarcities that are indicated by factor prices

Many labor market models have altered the Walrasian assumptions and it is important to remember that Bowles is not trying to change the Walrasian paradigm and/or his assumptions; he is merely stating the differences between his findings and those of Walrasian:

The existence of substantial employment rents in most jobs

• Can’t capture the income-related aspects of these rents by comparing the income levels of those with jobs and those without jobs because these groups differ in traits

• “Displaced workers suffer substantial periods of unemployment and that earnings on jobs held after displacement are substantially lower than pre-displacement earnings” (292)

• This has held true during times of strong labor demand

• One study illustrated that the subjective effect of joblessness was much larger than the subjective cost of the associated income loss

• Bowles Critique: The Walrasian model says that there may be jobs in which employees are indifferent to continuing in employment or having their employment terminated but this isn’t the case in reality because most employees prefer to remain employed

Real wages tend to vary with the level of employment

• Bowles Critique: According to the conventional Walrasian model, for a given capital stock, additional employment must lower the marginal product of labor → the wage

Employers devote substantial personnel and other resources to monitoring their employees’ effort levels

Labor effort appears to be quite variable and is very rarely ever subject to contract

Efficiency Wage Theory

Background: When wage is above equilibrium level then you have unemployment due to the fact that firms are offering fewer jobs at that higher wage level. Because of this more workers want to get jobs at that higher wage level and when workers seek jobs and can’t find them, it leads to unemployment

There are different ways to get the wage above equilibrium level such as the government requiring that firms pay a minimum wage above equilibrium level and workers organizing into labor unions and demanding a wage above equilibrium level

Definition of Efficiency wages: A wage that firms voluntarily choose to pay in order to increase profits. It’s the idea that a wage has 2 effects: it determines quantity of labor that’s traded in the market and wages influence the quality of labor that a firm hires. When a firm pays a higher wage it will get a higher quality of labor that makes it profitable for the firm to pay a wage above equilibrium level that can increase a firms profits. Efficiency wage is based on motivation and will lead to a more productive worker

There are many reasons as to why a firm would choose to pay a higher wage that is above the equilibrium level. Some reasons:

• Wage may influence quality of labor that the firm hires
• To give its workers a good deal, when you have a good deal you don’t want to lose that job
• Ex: Henry Ford: he decreased the workday and more than doubled the hourly rate of pay for a majority of his employees; paid his workforce a minimum of five dollars for an 8hr day. As a result of this, he had many people come to him seeking employment and profits and output per hour of production labor increased

Bowles Critique: This is inconsistent with the Walrasian view that profit maximization entails paying one’s employees a wage equal to their supply price. A happy, healthy worker is more productive and a firm that gives its workers more money than they can earn at another job leads to a happier, healthier worker. Higher wages also attract more skilled workers. In the Walrasian model, the employer is constrained by the workers’ labor supply decision and as a result the employer provides workplace amenities to lower costs of labor. A worker-friendly job will attract prospective employees at lower wages, the workers labor supply decision is the worker’s own utility, the employer will maximize profits by evaluating the importance of workplace amenities. However, this result does not hold when the effort is noncontractible

Equilibrium Selection Problem

The Walrasian paradigm fails to explain how institutions change and how preferences and beliefs of individuals co-evolve with their institutional environments. Indeed, it assumes that preferences and institutions are exogenous. In contrast, Bowles’ paradigm considers the economy as “embedded in a larger social and ecological system: co-evolving preferences and institutions.” (Bowles, 2003)

Some background

Institution

According to Bowles, institutions are "the laws, informal rules, and conventions that give a durable structure to social interactions, influencing who meets whom, to do what tasks, with what possible courses of action, and with what consequences of actions jointly taken." Bowles also proposes two processes that change institutions:

• The emergence within a society of a large number of individuals who act in ways that violate the convention, eventually displacing it (aka within-group tipping from one convention to another). This often takes the form of CONFLICT among interdependent actors who are differentially benefited by one or another institutional form

• Competition among groups governed by differing institutions (aka between-group competition). Thus, successful institutions are those that contribute to the survival of nations, firms, bands, etc. This also helps explain why market system proliferated throughout the world.

Modified Hawk-Dove Model

Now we will consider the modified version of the Hawk-Dove model mentioned in Section 2.3.2:

• Suppose we have n agents in this model.
• We want to randomly pair these agents to divide a good whose value is v.
• There are three available strategies for the agents in this model: grabbing (associated with grabber), sharing (associated with sharer), and punishing (associated with punisher). The agent’s strategy is not known before hand (aka it only reveals after an interaction).

Grabber plays the role of Hawk, Sharer plays the role of Dove, and we introduce a new agent here, Punisher. The mechanism between Grabber and Sharer is thus similar to that between Hawk and Dove. However, Punishers have a very interesting role. When Punisher meets Sharer or another Punisher, he will act like Sharer, i.e. split the good (each gets v/2). In contrast, when Punisher meets Grabber, all of the Punishers attempt to punish the Grabber.

If the Punishers successfully punish the Grabber, the good v is then distributed equally among all Punishers (so each Punisher gets v/n). If not successful, then the Punisher will bear the cost of defeat c (Note that c > v). It is obvious that the percentage of Punishers in the population is critical to whether they can defeat the Grabber.

Let β be the probability of successfully punishing a Grabber. Bowles simplified his simulation by assuming that this β is also the population frequency of Punishers (Thus, βn is the number of Punisher in the population). Hence, the expected payoff to a Punisher paired with a Grabber is:

∏(p,g) = β(v/βn) – (1- β)c = v/n – (1- β)c

We obtain the payoffs table in Figure 3.

Let α be the frequency of Sharers in the population. The average payoff for the whole system ∏* is

∏* = α∏_s + β∏_p + (1 – α – β)∏_g (Equation 1)

where

∏_s = (α + β) (½) v

∏_p = (α + β) (½) v + (1 – α – β) (βv – (1 – β)c)

∏_g = αv + β [ (1 – β)v – βc ] + (1 – α – β) (½) (v – c)

Another assumption here is that n is sufficiently large so realized payoffs are approximated by expected payoffs. We can represent this system using Figure 4.

Note that in Figure 4, the three line segments inside the triangle are perpendicular to each corresponding side of the triangle. They intersect at d. The lengths of these line segments represent the frequency of the strategy indicated at the vertex opposite the edge. For example, the frequency of Punishers is β. For every single point in this complex, these line segments sum to one. Therefore, we have homogeneous population at the three vertices.

The dynamics implied by Equation 1 is presented in Figure 5. Let γ be the frequency of Grabbers.

“The vectors indicate the direction of movement for a population composed by the frequencies given by the point at the base of the arrow.” For example, in Region I, the arrows for Sharers and Punishers point towards the corresponding edges, which means α and β are decreasing. The arrow for Grabber points away from the edge, which means γ is increasing. Note that Figure 5 also presents loci along which each of the population shares is stationary.

Hobbesian Equilibrium and Rousseauian Equilibrium

Hobbesian Equilibrium

Hobbesian equilibrium is characterized by frequent fighting over property, which has a low level of average payoffs.

In Figure 5, this equilibrium is represented by point b where β = 0, α = 1 – v/c, γ = v/c. This outcome is analogous to the equilibrium of the Hawk-Dove Game and is asymptotically stable. We can see that Punishers are not likely to invade this population since they do not do better when interacting with Sharers and will almost always lose to Grabbers when interacting with them.

Rousseauian Equilibrium

Rousseauian equilibrium combines “the unconditional sharing and collective upholding of social norms.” (Bowles, 2003) This equilibrium is represented by the left edge of Figure 5 along which α + β = 1. Observe that for the set of points on this edge where α < α_max (indicated by a), the equilibria there are “stationary but perturbations are not self-correcting.” These equilibria are known as Lyapunov (neutrally) stable equilibrium. Grabbers (or any mixed strategy with grabbing in its support) cannot invade this population since ∏_g < ∏_s = ∏_p (for α < α_max and α + β = 1). However, if non-best response behaviors sometimes occur, these equilibria are subject to drift downward along the edge of the simplex because Sharers and Punishers receive the same payoffs in the absence of Grabbers.

Equilibrium Selection

Equilibrium selection means giving reasons why one equilibrium is likely to emerge and persist, when other equilibria are possible. To choose an equilibrium, the model needs to take account of chance events.

Rousseauian equilibrium will not persist over long periods. Originally, given the large proportion of Punishers, all Grabbers, if appear, will be eliminated. However, after a while it will take more or less time for sufficiently many Sharers to accumulate so that Grabbers can now invade. The population will have drifted along the left-hand edge of the simplex in Figure 5, past point a, i.e. into the basin of attraction of the Hobbesian equilibrium. The Hobbesian equilibrium, however, is asymptotically stable.

Conclusion

In conclusion, preferences and institutions coevolve and exert influence on the development of each other. Both preferences and institutions are endogenous. With regards to endogenous preference, exogenous factors are the updating rule (copy behaviors that are prevalent and successful – which is also a distinct difference between Walrasian and Bowles’ paradigm) and a given structure of social interactions. With regards to endogenous institution, exogenous factors are preferences that influence which institutions are feasible and likely to persist, the distribution of behavioral types in a population which influences the equilibrium distribution of contracts, and the changes in technology (the rise of agriculture, the feasibility of food storage), which is important for promoting change in both institutions and preferences. (Marx’s view: “the development of the material forces of production” is one of the main dynamic forces in history)

The following table summarizes the contrasting approaches:

References

(1) Bowles, Samuel. Microeconomics: Behavior, Institutions, and Evolution. Princeton, NJ: Princeton University Press, 2006. Print.

(2) Gintis, Herbert. Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction. 1st ed. Princeton, NJ: Princeton University Press, 2000. Print.

(3) Gintis, Herbert. The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences. Princeton, NJ: Princeton University Press, 2009. Print.

(4) Hirshleifer, Jack. Price Theory and Applications. 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1980. 192-7. Print.

(5) Sandbec, Dix. "The Edgeworth Box beyond Laissez-faire." International Journal of Green Economics1.3 (2007): n. pag. Web. 13 Apr 2011. <http://dixsandbeck.ca/pdf/edgebox.pdf>.