Now that the example has been outlined we are able to translate this problem into the language of analytical game theory. Given the fact that the only players in this game are Jim and John, it is without question a two-person game. Also, since both players can unilaterally benefit from amicably dividing the estate prior to the 30 day deadline, this game is classified as non-zero sum. Successful division of the estate gives the brothers an additional 500,000 dollars than they would have otherwise had. Thus, in essence they are expanding the proverbial pie from 1.5 million to 2 million dollars. Also, the division of the estate between Jim and John is a one-shot game. Their father only dies once, and we are to assume that their father is the last living relative who intends to bequeath the brothers with an inheritance (for simplicity of the game we assume their mother died two years earlier and left all of her assets to her husband, Jack).
Although this is a one-shot game, the brothers may play it slightly differently than a typical one-shot game given their close relationship. This notion is proven by Robert Axelrod’s suggestions on how to promote cooperation in his book The Evolution of Cooperation. According to Axelrod “continued interaction is what makes it possible for cooperation based on reciprocity to be stable.” (Axelrod, 125) Furthermore, “mutual cooperation can be promoted …in three categories: making the future more important relative to the present; changing the payoffs to the players…and teaching the players values, facts, and skills that will promote cooperation.” (Axelrod, 126) The brothers are very likely to have future interactions, thus they are more likely to cooperate than a typical one-shot game in which you are likely to never interact with your opponent again. In this case defection could, and probably would forever strain the relationship between the brothers. During his life, Jack always stressed the importance of cooperation to his sons, and was always a strong proponent of equitable relations within his family. Now that their father has passed Jim and John hope to pass his cooperative mentality onto their children and keep it as an important family virtue.
With the guidelines of the game established, the observant student of game theory can see that this is a prisoner’s dilemma. It is in Jim’s best interest to choose to defect; worst case scenario he receives $750,000 and in his best case he can attain $1.125 million. Thus Jim’s dominant strategy is to defect. Conversely, Jim’s inferior strategy is to agree; his returns are $1 million and $375,000, both of which are lower than the respective defect payoffs. Similarly, John is faced with the exact same payoff matrix, thus, he will always defect because that is his dominant strategy. By agreeing, which is his inferior strategy; John runs the risk of only inheriting $375,000. Only one box in the payoff matrix includes both Jim and John’s dominant strategy: Box B2, which is the bottom right box in the payoff matrix. This outcome is known as the Pareto deficient Nash Equilibrium outcome.
To be considered a Pareto deficient Nash Equilibrium two axioms must be satisfied: neither player can benefit by changing strategy unilaterally, and both players can benefit by changing strategy bilaterally (in concert). Indeed, both of these axioms are satisfied in Box B2. If Jim chooses to agree, while John remains in his defect position, Jim will effectively lose $375,000 (750,000-375,000=375,000), while John will gain $375,000 (1,125,000-750,000=375,000). The same outcome occurs if John chooses to agree while Jim remains in his defect position.
On the other hand, if both Jim and John bilaterally choose to agree, then both of the brothers will be better off. By bilaterally agreeing, the brothers have successfully expanded the divisible assets of their father. They are eliminating their father’s charitable donation to the Eastern Surfing Association, and are keeping it for themselves. This outcome, represented in Box A1, is known as the cooperative strategy because it represents the optimal solution in the payoff matrix: the solution that makes all involved parties better off. Instead of splitting 1.5 million, they are now able to equitably divide 2 million dollars. To attain this solution, the brothers must both trust each other and remain confident that neither brother will try to undercut one another. If one brother agrees, while the other defects, the defector is able to increase their personal payoff $125,000, but in the process decrease their brother’s payoff by $625,000.
The aforementioned process of determining defecting and agreement strategies is more in accordance with analytical game theory, or the process of calculating machines with complete information playing the game to ‘win.’ In analytical game theory, the games are played by game theorists who are rational egoists, have complete information about the game, and have unlimited calculating ability. Analytically speaking, both brothers would defect to yield the greatest benefit to their individual interest, and forego the worst case scenario.
In reality, this game was played by two brothers who have a unique bond and love for one another that are incomprehensible to a calculating rational game theorist. As Professor Macy pointed out in lecture, individual self-interest is empirically variable, not axiomatic; there are no axioms to describe the bonds between two brothers. Further, cooperation is largely based on concern for the welfare of others. Thus, evolutionary speaking, both brothers would agree to cooperate because they truly are concerned about the welfare of one another. Mutual agreement allows both brothers to be better off; in a familial setting making other people happy, in particular your sibling, is often priceless.
Though, it must not be forgotten that evolutionary game theory is a rule-based model; there are cognitive constraints that limit the amount of information available. Considering this, mutual agreement may in fact be the utopian solution, although even in evolutionary game theory this result is not guaranteed. In evolutionary game theory, the players in the game look ahead to the future by actions that took place in the past. Brothers, we can assume, have a strong history of interaction and by looking back on previous events it is often possible to predict the future of their relationship. Thus repetitious actions, not analytical calculations, are likely to determine the payoffs in this inheritance game. It’s completely possible that Jim may indeed try to influence John to defect. If Jim then agrees to negotiate, he limits John’s inheritance and maximizes his personal inheritance. If Jim has a history of such devious actions, then John is more likely to play the game with extreme caution. The relationship between the brothers is the key ingredient in determining which solution the brothers will ultimately end up with.
There are other factors that also determine the ultimate solution. Earlier it was noted that Jack always stressed cooperation within his family. Thus, the rule that Jack set forth in his will was for his sons to cooperate in order to attain the maximum inheritance possible. It’s possible that the $500,000 designated for the Eastern Surfing Association was intended to be nothing more than a weak threat included in the will to foster cooperation between the brothers. Now that their former mediator (their father) is gone Jack wanted to leave the brothers with a memory of how his goal was to promote a cooperative relationship between his sons. This notion is known as kin altruism, which promotes cooperative behavior within biological units (families) or close cultural relatives. By proving to his sons that their inheritance could be expanded with cooperative effort, Jack was able to achieve his goal of kin altruism.
Also, the notion of reciprocal altruism might have played a key role in the decision making process. Reciprocal altruism refers to the exchange of cooperative acts within a relationship. By coming to a mutually beneficial agreement with regard to their inheritance, the brothers are likely to exchange kind acts in the future as a way of showing their gratitude for one another. This scenario can be thought of as Tit-for-Tat. The first cooperative act of expanding and equally dividing their inheritance can be though of as the initial ‘tit;’ this reciprocating process of cooperation will continue until either Jim or John decides to defect.
Even if Jack’s will seems overly detailed and complex, it serves to reinforce the benefits of cooperative negotiations. If you look at the game from the perspective of analytical game theory, the brothers will both choose to defect and effectively mistrust each other out of an extra $250,000. On the other hand if you look at the game from an evolutionary standpoint, the brothers’ previous interactions and past repetitious behavior, will likely lead to a mutually beneficial outcome in which the inheritance is expanded and each brother inherits $1 million. From the evolutionary standpoint this game is in fact a one-shot game, however the reasoning behind the cooperative efforts of both brothers is more representative of a repeated game. Although the brothers might not have any future negotiations, or games, they are sure to have future interactions and dealings with one another. Thus, they have a vested interest to mutually cooperate to expand the inheritance.
As you can now see, problems arise when each sibling enters into a negotiation with a mindset of greed and a desire to inherit as many assets as possible. Analytical game theory falls short in providing a real solution to the game. The calculating analytical game theorist worries more about defection than collaborating to reach a mutually beneficial solution. Evolutionary game theory, however, is able to provide a real-life solution to the game. The evolutionary game theorist plays the game according to rules, which will guide the player to seek a mutually beneficial solution.
In the end Jim and John decided to ‘enter into negotiations’ to settle their father’s estate. Without the assistance of a mediator, a lawyer, or even their wives, the brothers each decided to take $1 million. The bond between brothers is outside the comprehension of the analytical game theorist.
