Game Theory and the Kingdom of God (A Quirky Series Installment), Part 11: "Cutting Cakes, Fairness and the Ultimatum Game"

I want to make some important points about my last post, the victory of Tit for Tat, but before I do I want to talk about another kind of game that I will discuss along with Tit for Tat to make some observations about the Kingdom of God.

I want to talk about the Ultimatum Game, but let's start with cakes.

Cake cutting algorithms have long fascinated recreational mathematicians, and some professional ones as well. The classic cake cutting problem is this:

You have a cake that needs to be fairly divided between two people. Both people are self-interested in that each wants the biggest portion they can obtain. Can you identify a procedure for them to divide the cake that will ensure a fair division?

I pose this problem to my undergraduates and it doesn't take them too long to find the solution:

1. Have Person A make a cut.
2. Have Person B choose between the two pieces.

This procedure (i.e., algorithm) is assured to generate a fair division in that the self-interest of the players is harnessed to create fairness. Person A, who gets to make the cut, will not be the first to choose. Knowing Person B will select the larger of the two pieces after the cut, Person A will make the most even cut she can. After Person B chooses, if Person A doesn't like her piece she has no one to blame but herself. After all, she is the one who made the cut...

The 2-person cake cutting problem is, in the delightful parlance of mathematicians, trivial. What is less trivial is a 3-person cake cutting procedure. Then the 4-person. Then the 5-person. And so on. Given that mathematicians seek generality, what many look for is the N-person procedure, a generic procedure that can be used to ensure fair division from two people to two million.

All this is background for a game called the Ultimatum Game. The Ultimatum Game was invented by Werner Guth of Humboldt University in Berlin about 25 years ago. The Ultimatum Game has since begun to attract the attention of scores of social scientists and behavioral economists for what it is teaching us about human psychology and social relations.
Here is how the Ultimatum Game, reminiscent of the cake cutting problem, is played:

Two players have to divide $100.

Player A proposes a division (e.g., "$50 for me and $50 for you" or "$75 for me and $25 for you" or any other division).

Player B after hearing the proposed division has two choices:

1. Accept the division. If accepted, the players split the money as specified by the division taking home the money they were allotted.

2. Reject the division. If rejected, neither Player A or Player B get any money. Both go home with nothing.

Simple game, huh? Amazingly, however, this simple little game is shaking the foundations of classic economic theory.

Classic economic theory works with a model, Homo-economicus, "Economic Man." Economic Man is an idealized decision-maker, the person whose behavior classical economic theory attempts to predict with its rigorous mathematical equations. Some assumptions govern the behavior of Economic Man. Mainly, Economic Man is assumed to be a rational utility-maximizer. That is, given two choices Economic Man will behave in a "rational" manner by making the choice that will yield the greatest satisfaction (see my first post in this series). This assumption seems perfectly reasonable. But laboratory observations of real people in real interactions, such as the Ultimatum Game, are challenging this assumption. Perhaps Economic Man is not a good model for real, flesh-and-blood people.

To understand this, let's look at what economists would call the "rational strategies" Players A and B should follow in the Ultimatum Game. First, if Player A were rational, he should propose a very lopsided division, like "$95 for me and $5 for you." Why? Because Player B is assumed to be rational. That is, despite the unfair division proposed by Player A, Player B's choice is simple: Do I accept and go home with $5 or reject and go home with $0? Given the choice between $5 and $0 what would Economic Man choose? Economic Man, behaving rationally, goes with the $5.

But here is the problem. In the laboratory, people in the position of Player B, when presented with very lopsided offers, don't behave rationally. They get rather upset and reject the offer. Think about it. You're sitting with a stranger who can propose a nice, even, 50/50 division where both of you can walk out of the laboratory each with a cool $50 in your pocket. Instead, your partner (in the position of Player A) looks at you and says, "I'll take $99 and you can have $1." What would you do in this situation? Well, most of us would say, "Hey, that is greedy and unfair! So, no way you walk away with $99! I reject the division."

Interestingly, people in the Player A position seem to know this is coming. So, rather than assuming they are playing with "rational" players and, thereby, proposing very lopsided divisions, most players in the Player A position propose divisions close to 50/50. It seems that the people proposing the division know already they are playing with "emotional" rather than "rational" players. That is, many of us, when faced with an "unfair" division, would cut off our nose to spite our face.

Overall, then, in the Ultimatum Game most proposed divisions are close to 50/50. And these divisions are readily accepted. By contrast, lopsided divisions are not often proposed but, when they are, they are rejected. In sum, both in the proposing and in the accepting/rejecting normal people do not behave rationally. We are not, sad to say (if your are a classical economist), images of Economic Man.

What does all this have to do with the Kingdom of God and Tit for Tat? Lots. I'll get to all that tomorrow. But to give you some payoff for reading such a long post, let me give you peep at the topic for tomorrow...

The Ultimatum Game demonstrates that humans do have a "logic" of how relationships and social interactions are to be managed. And, as we have seen, it is not the logic of economics. Rather, it is a much cruder and simpler logic. A very powerful logic. It is the logic of fairness and reciprocity. The Ultimatum game illustrates the logic of fairness, we "irrationally" demand a "fair" division and will accept nothing less. Tit for Tat, although a computer program, nicely captures the reciprocity dominant in human relations, both the good (e.g., "You scratch my back and I'll scratch yours") and the bad (e.g., "An eye for an eye").

These mental biases for fairness and reciprocity are universal features of human nature, manifested across all cultures. Consequently, these biases form the foundation of humanity's innate moral psychology. And, as a start down the road of ethical living, fairness and reciprocity are not bad first steps.

But, although these first steps come naturally for humans, they set the bar too low. For citizens in the Kingdom of God, we are called to something higher. The question is, do we get there? Or, do we in the church reduce Kingdom living to simple reciprocity, attempting to keep our relations with others "balanced"?

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3 thoughts on “Game Theory and the Kingdom of God (A Quirky Series Installment), Part 11: "Cutting Cakes, Fairness and the Ultimatum Game"”

  1. Re: The Ultimatum Game.

    Surely the laboratory behaviour is not irrational. When offered $1, person B disdains to accept because they value the pleasure of punishing miserly person A by denying them their $99 more than the measly dollar that person B stands to gain from person A's proposal.

    If the pot was $1 million, and person A offered $10,000 (i.e. in the same proportions) the above logic would not apply, and surely the overwhelming majority of persons B would accept person A's proposal?

    i.e. people are motivated by value, which may or may not be financial.


    He who knows nothing of game theory.

  2. 'He who knows nothing...' again here.

    So I suppose a further question might be... from whence comes that pleasure of denying player A their $99? Is it a evolutionary social consciousness mechanism to punish the transgressor of a fairness taboo? Is it the first step in an unconscious tit-for-tat strategy? Both??

  3. It's more than the 'satisfaction' of punishing an unfair proposal. People are acting as if this is not the last iteration of this game. There will be future iterations of this game, likely with other opponents. If they give in to the 95/5 proposition, others will walk all over them too. But if they stand tall and reject the unfair offer, the next proposal will be much more fair. It's not much different than the repeated PD game.

    Taken in isolation, it's always worthwhile for the US to give in to terrorist demands. Saving the lives of a plane load of Americans is worth a few million bucks and a free flight to Libya. (The 95/5 offer.)

    But refusing to give in to terrorists demonstrates an overall strategy. It's more costly in the short run, but very rational in the long run because it deters the unfair proposal.

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