Game theory is the science of strategic reasoning, in such a way that it studies the behaviour of rational game players who are trying to maximise their utility, profits, gains, etc., in interaction with other players, and therefore in a context of strategic interdependence.

Historical Framework

The initial game theory discussions and analysis can be traced back a long time before the 20^th century, with works from authors such as James Waldegrave and his minimax mixed strategy solution for a two-person game in 1713, James Madison and his theoretical analysis of what would the effect of different tax systems be, or Antoine Cournot’s solution to a duopoly that resembles to what would later be known as Nash equilibria. However, it is not until the 20^th century, that game theory is broadly developed. We can differentiate several periods in the game theory evolution through the 20^th century. However, many concepts were continuously developed through the years.

During the earliest years of the century, from 1910 to 1930, the main focus of game theory was on  strictly competitive games commonly referred to as two-person zero-sum games. In this kind of games, the preferences (and payoffs) of a player will always be opposite to the other player’s and therefore cooperation between both players will be pointless. This kind of games has been extremely productive, as they have set the bases for future development of game theory, becoming its first main milestones. During these years, John von Neumann’s contribution was especially important, and thus, he is considered as one of the fathers of game theory.  In his “On the Theory of Parlor Games”, 1928, he introduced concepts such as the extensive (or tree) form for the description of sequential games and the minimax theorem, providing empirical support to everything he wrote. Another important contribution during these years is the introduction of the strategic form (game matrix) of a game that represents each player’s profile in a matrix form.

The following 20 years were heavily influenced by von Neumann and Oskar Morgenstern’s work, which would culminate in the joint publication of their book “Theory of Games and Economic Behaviour”, 1944.  With this book, game theory gained the status of an independent scientific discipline. One of the major contributions of this book is the application of strategy game concepts instead of random ones, the introduction of a notion of a cooperative game, its coalitional form, and the ‘Neumann-Morgenstern stable sets’. They were the first to extensively apply game theory for practical reasons, especially for analysing economic behaviour. It is also important to stress that the impact of this book has gone beyond its own time.  Many future developments emerged from it, such as the notion of core and transferable utility, and the further development of expected utility theory. Other developments from this period include games with continuum of pure strategies, the computation of minimax strategies and advances in mathematical methods that would be instrumental to later work.

The 1950s were a key period for game theory. The initial phase of incubation had reached its end and the economic perspective and application took the lead, overtaking other perspectives such as the military one. It was during this period when John Nash laid the essential groundwork that would result in the general non-cooperative theory and for cooperative bargaining theory; Lloyd Shapley was another key figure as he defined the value for coalitional games (cooperative games), he initiated the theory of stochastic games, co-invented the core with D.B. Gillies, and, along with John Milnor, developed the first game models with a continuum of players; Harold Kuhn worked on behaviour strategies and perfect recall. The prisoner’s dilemma, which is probably the best known game, was formalized by Al Tucker also during this period. It was also during these years, when the Folk theorem appeared as a result of the interest and study of repeated games, used to analyse punishment strategies in collusion agreements.

The period from 1960 to 1970 was also important in the development of game theory. The discipline expanded, not only theoretically, but also geographically. With the extension of games such as those with incomplete information and non-transferable utility coalitional games, game theory became more widely applicable, and new research centres were established outside the U. S.

From 1970 on, game theory acquired maturity. Up till then, most of game theory concepts and discoveries had been spread word to mouth. However, during this period economic theory journals published many articles related with game theory and even, new journals entirely dedicated to game theory became popular. The interest and study in politics and political economic models grew. The use of non-cooperative game theory to a large variety of economic models brought also the study and refinement of the equilibrium concept. Other advances were made in other areas of research such as complete and incomplete information repeated games, stochastic games, value, core, nucleolus bargaining theory, games with many players and etc. The use of game theory proved to be useful in a wide diversity of areas that include biology, computer science, moral philosophy and cost allocation, to name a few.

It is worth to note, that game theory should be regarded as a tool to find where incentives will lead, but it makes no moral recommendation to what choices should be taken. This science could be pictured as the study of selfishness, but without recommending it.  Welfare economics and its main tools (such as Pareto optimality and compensation criteria) study these problems outside the selfishness condition.

The theory of consumer choice under situations of risk and uncertainty belongs to the field of microeconomics. Risk and uncertainty are sometimes interchangeable terms but their meaning is easily misunderstood. Frank Knight in his “Risk, Uncertainty and Profit” 1921, treated this subject and posed a fundamental distinction between the two, formulating the definition that, ever since, became the most widely used.

In a situation that involves uncertainty the outcome is unknown and agents cannot (or will not) assign probabilities to each outcome. Sometimes it is said that uncertainty is an unknown-unknown, while risk is a known-unknown, since agents assign probabilities to each outcome.

Consumer choice under uncertainty is studied mainly in game theory, while risk is usually analysed using the expected utility theory approach.

Prospect theory belongs to behavioural economics and outstands as an alternative model to expected utility theory, as the neoclassical assumption of the rational agent is put into question. This theory was developed by Nobel laureate Daniel Kahneman and his collaborator Amos Tversky in their “Prospect Theory: An Analysis of Decision under Risk”, 1979. They used results they gathered both from their own empirical observations and from various experiments.

Individuals establish preferences depending on a certain situation and specific circumstances, rather than in absolute terms. This means that depending on their initial situation, agents will act one way or another. One of the outcomes of this reasoning leads to behavioural asymmetries between situations of potential loss or gain. Individuals, for example, are generally more adverse to losses than they are attracted to gains. An endowment effect has also been distinguished, as the compensation required by someone to give away a good, is larger than he would be willing to pay in order to acquire it.

Prospect theory postulates the existence of a value function V (Δw) that depends on the deviation from a reference point (determined in terms of wealth), and a function ∏ (p_i) that turns the weighted probabilities applied in decision-making. In this context A is preferred to B if: ∑∏(p_A) V (Δw_A) > ∑∏(q_B) V (Δw_B)

Function ∏ (p) is defined on the interval [0, 1] transforms the probabilities into decision making weighted probabilities. It is a function characterized by being monotonically increasing that intersects the 45 degrees line near the origin, so that the weights in decision-making are smaller than the probabilities, except for the very low probabilities.

The value function V (Δw), which is shown in the adjacent diagram, is dependent on changes in wealth relative to a reference situation, having a sigmoid-shape: being concave for gains (risk aversion) and convex for losses (risk seeking). It presents a discontinuity at the origin with a steeper slope for losses than for gains.

This theory helps understand why the same individual can be, at different situations, risk-avoiding or risk-seeking. It explains behaviours observed in the economy such as the disposition effector why the same person may buy both a lottery ticket and an insurance policy.

After John von Neumann and Oskar Morgenstern developed the expected utility theory in their “Theory of Games and Economic Behaviour”, 1944, various different approaches were developed. Although the expected utility function helps us understand the real world, it is important to remember that it is only a simplification of it. Expected utility theory does not completely reflect how agents interact in the real world. Furthermore, agents’ behaviour in the real world seems to systematically break some of the axioms. Let’s see some of the alternative approaches that where formulated after the original expected utility theory came out.

Friedman-Savage perspective:

Milton Friedman along with Leonard J. Savage, developed in their “Utility Analysis of Choices Involving risk”, 1948, their own utility function known as the Friedman-Savage utility function. They argued that a single individual could have different utility functions depending on their initial wealth. The implication of an individual being, at the same time, risk-loving and averse, implies that its utility function has different curvatures as shown in the following figure.

Markowitz’s perspective:

Although the Friedman-Savage perspective somehow explained the flaws in the original expected utility theory axiomatic, the critiques and alternative theories didn’t stop there. Harry Markowitz, who was a student of Milton Friedman, criticized the Friedman-Savage utility function. Markowitz argued in his paper “The Utility of Wealth”, 1952, that the final concavity of their function assumes that individuals with the highest incomes would never gamble. As a prelude to Kahneman and Tversky’s prospect theory, he proposed measuring utility based on a reference level instead of in absolute values. This implied that, to individuals, small gains would provide an increasing utility, while big gains would provide a decreasing utility, as shown in the following figure.

The expected utility theory deals with the analysis of situations where individuals must make a decision without knowing which outcomes may result from that decision, this is, decision making under uncertainty. These individuals will choose the act that will result in the highest expected utility, being this the sum of the products of probability and utility over all possible outcomes. The decision made will also depend on the agent’s risk aversion and the utility of other agents.

The base of the theory are lotteries, or gambles, (L_n) each one defined by all possible outcomes or consequences (C₁,C₂,…,C_n) and their corresponding probabilities (p₁, p₂,…,p_i, with ∑p_i=1). 

EU(L) = U(c₂)p₁ + U(c₂)p₂ + … + U(c_n)p_n

The term expected utility was first introduced by Daniel Bernoulli who used it to solve the St. Petersburg paradox, as the expected value was not sufficient for its resolution.  However, John von Neumann and Oskar Morgenstern, in their book “Theory of Games and Economic Behavior”, 1944, considered the cornerstone of expected utility theory, provided great contributions and built a mathematical foundation for Bernoulli’s solution of the paradox. They developed a set of axioms for the preferential relations in order to guarantee that the utility function is well-behaved:

1.Axioms of order:

Completeness L₀L₁ or L₁L₀ or L₀∿L₁

Reflexive L₀L₀

Transitive L₀L₁ ; L₁L₂ ↔ L₀L₂

2.Continuity: L₀L₁ ; L₁L₂

with p∊ [0,1] so that L₁∿ p L₀+(1-p)L₂

3.Rationality: in order to maximize results, the highest probability will be chosen (ceteris paribus)

4.Rational Equivalence: agents will evaluate rationally the probabilities of the different results

5.Independence (sure thing principle): if L₀L₁ →β*L₀+ (1- β)*L₂ β*L₁ +(1- β)*L₂

Oskar Morgenstern and John von Neumann’s expected utility theory, which analyses individuals’ risk aversion, proves that different individuals have different perspective towards risk. Risk averse individuals have, by definition, a greater preference to avoid risky situations than risk-loving individuals and, to this end, they will be willing to pay an extra amount of money in order to mitigate (or eliminate) the bad consequences of such a risk.

Expected utility theory holds that the demand for insurance can be translated as a demand for certainty. Individuals will prefer to buy insurance in order to assure a certain amount of money (or to have a guarantee of lower losses), instead of its actuarial equivalent uncertain one.  The insurance market allows agents to cover themselves against risk. Insurance companies take advantage of risk averse individuals to charge an extra surcharge to pay costs which are not covered by the premium. How individuals perceive insurances depends on their prices, and on the individuals’ preferences and budget constrain. A risk averse individual may be willing to assure against a potential loss, but will pay only up to a certain price for this insurance: if the price exceeds this amount he will not acquire the insurance.

We will use an analytical example for a better understanding:

Imagine an individual with an initial wealth of W₀ who faces the possibility of getting robbed an amount of R. He or she has the option of insuring an amount of K for a risk premium of λK. There are two possible scenarios:

S₁, with a probability of happening p₁: there is indeed a robbery, the individual loses R;

S₂, with a probability of happening p₂: there is no robbery.

We’ll have a budget constraint, in terms of wealth, such as:

The optimum point, given the budget constraint and being MU the marginal utility, is:

Attitudes and behaviour towards risks have been, and still are, highly studied fields in psychology and their economic applications have been meaningful and of high importance. While some may be willing to assume risks in order to gain economic profits, others will prefer to avoid those risks, even though they may report higher expected payoffs. The first type of individuals is considered risk loving individuals, while the second ones are risk averse individuals.

The expected utility function helps us understand levels of risk aversion in a mathematical way:

Although expected utility is a term coined by Daniel Bernoulli in the 18^th century, it was John von Neumann and Oskar Morgenstern who, in their book “Theory of Games and Economic Behavior”, 1944, developed a more scientific analysis of risk aversion, nowadays known as expected utility theory. The analysis of risk aversion we are about to see is based on their work.

In order to shape an individual’s expected utility function, the different prices (payoffs) need to be ordered. After assigning random numbers to the upper and lower levels (here 0,1), intermediate values are considered asking the individual what is the probability of a given outcome so that he or she would be indifferent whether to play or to take a guaranteed prize. This has to be repeated until the function is obtained.

The expected utility function has a set of important properties:

-it increases with income;

-it is additively separable;

-it is bounded between an lower (0) and a upper limit (1);

-it is differentiable.

Now, let’s define a few terms we’ll need in order to analyse any individual’s degree of risk aversion:

-Expected value, also referred to as mathematical expectation, which is the expected payoff :

-Utility of the expected value:

-Certainty equivalent, which is the certain payoff that reports as much utility as the utility of the expected value: 

-Expected utility, being such that it is equal for the certainty equivalent and the expected value:

Risk aversion (green) may imply that an individual may refuse to play a fair game even though the game’s expected value is zero. While on the other hand, risk loving individuals (red) may choose to play the same fair game. In case of risk neutral individuals (blue), they are indifferent between playing or not. The utility function for each case can be graphically drawn (being RP the risk premium and A the Arrow-Pratt measure of absolute risk aversion):

The theory of consumer choice under situations of risk and uncertainty belongs to the field of microeconomics. Risk and uncertainty are sometimes interchangeable terms but their meaning is easily misunderstood. Frank Knight in his “Risk, Uncertainty and Profit” 1921, treated this subject and posed a fundamental distinction between the two, formulating the definition that, ever since, became the most widely used.

In a situation that involves risk the outcome is unknown, but the distribution or the probability of its occurrence is known. These probabilities can either be objectively specified or subjectively, meaning that the agents will form their expectations regarding these probabilities. Sometimes it is said that risk is a known-unknown while uncertainty an unknown-unknown, since in the latter agents cannot (or will not) assign probabilities to each outcome.

Consumer choice under risk is usually analysed using the expected utility theory approach, while uncertainty is studied mainly in game theory. However, although different models have been developed for both situations, risk situations, are the ones that have received the most attention and economists have concentrated on the analysis of risk probably due to its well-defined probability distribution in comparison with the random variables that uncertainty involves.

It must be mentioned that Daniel Kahneman co-authored with Amos Tversky in 1979 the paper “Prospect Theory: An Analysis of Decision under risk”, which drastically changed the way economists analyse decisions under risk. Their analysis is known as prospect theory, which studies the results from a series of empirical data and has very different conclusions from those drawn by expected utility theory.