A Game Theoretic Exploration of a Consumers Decision Making in the Market for Life Insurance - Report Example

A game theoretic exploration of a consumer’s decision making in the market for life insurance Introduction Any individual consumer purchasing insurance or exhibiting an intention to do so is essentially in the process of ensuring protection against possible undesired outcomes in future. Under standard assumptions of rationality, a consumer is an agent whose objective is to maximize utility. In the market for insurance, the buying decision is based upon the same utility maximization principle, though under conditions of uncertainty. In the presence of uncertainty, the consumer is not certain about any outcome and is assumed to be able to attach subjective beliefs to the possibility of occurrence of each out come and thereon aims to maximize his/her expected utility subject to the budget constraint (Varian, 1997). So, any theoretical depiction of a consumer’s buying decision in the insurance market has to be based upon this maximization of expected utility. In the present endeavour, we first introduce the basic principles of game theory and thereon proceed to apply them to model the given problem. 1. 1 A brief introduction to the relevant game theoretic notions Game theory is a mathematical tool that attempts to model situations of cooperation and conflict (Kreps, 1990). Game theoretic analysis assumes the participating agents to be intelligent as well as rational, implying the agents to be able to infer anything that we, the students of the game are. Alternatively, intelligence implies that if we can decide that a particular strategic choice is in the best interest of any particular agent given the strategy choice of others, then so are all the players of the game able to do (Kreps, 1990). A game is of the cooperative or non-cooperative type depending upon the nature of the players’ interests in whether they are conflicting or not. Given the type of the game, any game is defined by primarily by its environment, which is a defined concept in theoretical terms. By environment of a game, we refer to the number of players, the strategy set and the timing of decisions. Along with this, the other important factors are the nature of informational presence in the game. A game is termed as one of perfect and complete information if, all information related to the game is common knowledge, and all past actions are observable at each point of the game. Here by common knowledge we refer to information being available to each player, and each player being aware of this availability. For two players A and B this implies that some information is common knowledge if player A knows something and player B knows it as well and moreover player B knows that player A knows this and player B knows that player A knows that player A is aware of this information, and so on ad infinitum. 1.2 Nash Equilibrium The basic solution concept that shall be applied to the game is going to be Nash equilibrium. So, it is imperative to introduce the notion of Nash equilibrium. Intensive technicalities shall be avoided and in what follows, Nash equilibrium shall be introduced as much as possible as an intuitive concept. Nash equilibrium in essence refers to a situation where the best responses or best possible strategies of the involved agents match so that unilateral deviations are not gainful (Binmore, 1999). In other words, a given combination of choices can be said to be Nash equilibrium if given the other players choices, moving away to any other choice is does not offer higher benefits to any players. So, in a game with two players A and B, (S*A, S*B) is the Nash equilibrium strategy profile if from a possible set of strategies, S = {SA, SB} [where Si represents the strategy set for the ith player], given that player B plays S*B playing anything other than S*A is not gainful for player A and given that player A chooses S*A, Playing anything other than S*B is not gainful for player B. Armed with the primary ideas about the basic defining properties and natures of games and the solution concept to be applied, we now further our discussion by initiating the process of developing the given situation of a consumer. It will be attempted to keep the game and the ensuing analysis as simple as possible. 2. The Model To keep the analysis simple we shall assume the simplest of structures possible to envisage the situation. Following are the definitive assumptions specific to the model: 2.1 The players Let us start by first defining the environment of the game. We shall consider two players, the rational and intelligent consumer and Mother Nature. The individual is uncertain about his future, there are two possible outcomes one of which is undesired, and there is a positive probability P of the undesired out come which he/she wants to insure against. Let, Y1 represent the monetary equivalent of the outcome which is more preferred with Y2 being the monetary equivalent of the undesired outcome, and we define U(Y) as the consumers preference function with U’(Y) > 0 and U’’(Y) < 0 => U (Y1) > U (Y2). The consumer has the option of purchasing insurance for which a premium of the amount ‘D’ has to be paid and a lump sum income ‘I’ is paid in the case that the undesired outcome materializes. Considering the purchase of insurance will be rational only if U (Y2) < U (Y2 – D + I). If we consider that the undesired outcome is loss of life, then we can think that it is the certainty of an income being delivered to the individual’s family in the unfortunate event of his death that compensates his loss of satisfaction from the notion of possible death. Mother Nature determines which of the outcomes is to materialize. We assume that she is indifferent between the two possible outcomes so that her payoff from either outcome is zero. 2.2 Timing of decisions We assume that the individual moves first, deciding whether to buy insurance or not and after that nature plays in deciding which outcome materializes. Initially we ignore the probability P. 2.3 Extensive form representation So, we can represent this situation in the form of the following game tree: For expositional advantage we assume the following values for the defined variables: Y2 = - 3, Y1 = 2, D = 1, I = 4 = > (Y2 – D + I) = - 1, (Y1 – D) = 1 To visualize the solution it is best to represent this in terms of the following payoff matrix with the assumed numerical values1: 2.4 Solution Here the strategy choices (Buy, cause accident) and (Not Buy, Not cause accident) are the Nash equilibrium solutions. Consider first the strategy combination (Buy, cause accident). Given that nature chooses to cause the accident, the individual will be hurt by a loss of 3 units if he chooses not to buy insurance, and thus does not gain from deviating from ‘Buy’. Given that the consumer decides to buy insurance, nature does not gain by deviating from ‘cause accident’ as we have assumed indifference on her part from the two outcomes. Similarly for the second strategy combination (Not Buy, Not cause accident), if nature chooses not to cause the accident, then the consumer looses by a single unit of income if he chooses to deviate from the ‘not buy’ choice. Again, if the consumer chooses not to buy, nature remains indifferent between choosing to cause or not to cause the accident. Now we proceed to show why the other two strategy combinations are not Nash equilibrium. First consider the combination (Buy, Not cause accident). Though the individual gains from the situation of nature choosing not to cause the accident if he chooses to buy insurance, it is not the best response if nature’s choice is not to cause the accident as, if nature chooses not to cause the accident, then the consumer gains from choosing not to buy. So a deviation on the part of the consumer is beneficial and thus this combination can not survive as Nash equilibrium. Again, in case of the remaining possible strategy combination, (Not Buy, Cause accident), given that nature chooses to cause accident, the consumer gains from choosing to insure rather than proceeding without any insurance coverage. 3. Conclusions So what emerges is in this simple setup, (Buy, cause accident) and (Not Buy, Not cause accident) emerge as the two Nash equilibrium strategy combinations. Now, consider the role of the probability of occurrence of the accident, P as believed by the individual. If, P = 1, that is the individual was certain that the accident was to occur, he would choose to have his life insured. Again if P = 0, implying that he was sure that the accident would not occur, he would not choose to insure his life. Thus, there must exist a critical value of the probability P (say P*) lying between zero and unity for which the individual shall be indifferent between the two choices of buying or not buying insurance coverage. So, the consumer shall make the purchase for all P> P* and choose not to buy for any P < P* This has an important implication for Insurance companies. If this critical P* can be identified for any given consumer that insurance is bought by a consumer can be ensured if it can be impressed upon him that the probability of occurrence of the undesirable event is greater than this critical subjective value. References: Binmore, K., (1999) Fun and Games: A text on game theory, Houghton Mifflin Company, USA Kreps, D. M. (1990) A course in Microeconomic Theory, Prentice Hall International, UK Varian, H.R., (1997) Intermediate Microeconomics, W.W. Norton and Company, New York Read More