2: I Cut and You Choose

The notion of fair play is fairly common to most human cultures, and there is even evidence that it exists within the animal kingdom (the author cites and experiment in which capuchin monkeys became upset when others received a better reward for performing the same task). Fairness is not to be categorized as a concern for the other party's welfare, but for one's own: each person wants to get their own fair share.

When dealing with finite resources, a strategy for mitigating conflict is "I cut you choose" - the example being two children who share a piece of cake: one of them gets to determine where the cake is cut, the other gets to choose. Each is then responsible for an equal outcome. The cutter can't complain because he assumed the responsibility of making an equal distribution, and the chooser can't complain because he was the one to choose his portion. The author uses the term "minimax" to describe the cutter's strategy: he arranges the situation as to minimize his maximum loss, regardless of the decision made by the other party.

The insurance industry is based on the concept of minimax: considering the high cost of a possible loss, the customer chooses to take a small lost (paying an insurance premium and getting no benefit from it) to avoid the possibility of a much larger one (in the event of a catastrophe or accident).

The strategy has also been put into use in international law - the UN convention for mining in international waters calls for a nation to define an area it wishes to exploit and divide it into two portions - but an independent agency representing other (developing) nations would then choose one of the tracts for its own future use.

The drawback to this strategy is that it is applicable only to zero-sum situations in which the gains and losses balance out neatly. Such situations are rare in real life

Another drawback is that value is subjective, and individuals may have very different values. In the case of a divorce, a piece of jewelry may have a low value to one player (scrap value) but a high value to another (sentimental value), and the first may exploit the second's perception of value to gain more for himself.

Also, the strategy depends on the presence of an independent authority to stop each party from trying to get more than a "fair" share by cheating, deception, or intimidation.

Different Strokes for Different Folks

The author tells an anecdote of a situation where he was at a party where a cake was sliced and shared out - and the last two pieces were left to himself and another person. He decided to let the other person choose, and they chose the smaller piece. When asked why they chose the smaller of the two, they remarked they would have "felt bad" about having been greedy.

The author notes that this might not have been entirely true - it may be that it was less about emotions and more about social embarrassment (other people will think less of them), but regardless of their true motivation, it illustrates that game theory has overlooked a significant factor: the "reward" in the situation is not limited to a larger bit of cake, but it also entails the emotional or social outcome of the choice.

Thus, in choosing the smaller slice, the other guest sought the greatest benefit to herself: her emotional contentment and social esteem being more valuable than a few milligrams of additional cake.

The notion of utility comes into play with price differences in the retail market. Customers will be willing to pay more for an "identical" good at a corner grocery than at a supermarket some distance away, because they gain convenience. In that way, you can roughly assess the dollar-value of convenience as the mark-up customers are willing to pay to avoid the trip to a location where a good can be had at a lower price.

Dollar-values can also be placed on intangible benefits that are otherwise hard to quantify. A few examples are provided of a town paying farmers to leave hedges that made the countryside more appealing to tourists (the "value" of the scenery being monetized in this payment), or a wildlife preservation society purchasing land to prevent it from being leveled for development (the value of the habitat being derived from the price they were willing to pay).

It seems that such a strategy could be well put to use to solve common problems: to stop deforestation of the Brazilian rainforests, simply buy the land, or pay the farmers to leave it as it is (in an amount that would compensate them for the profit they would have earned by clearing and farming it).

The problem is that this practice approaches the notion of bribery, and tends to create an expectation. If a parent offers a child a reward to clean his room, the child will expect the reward every time he does this task, or will refuse to do it if not paid off - when in reality, it is a task the child is expected to do anyway, but the exasperated parent resorted to offering a reward "just this once" to see it done.

However, bribery is entirely acceptable within game theory, and is often used in real life - though it may be called an "incentive" or a "reward," it comes down to the same thing: it is a method of getting a person to enter into an agreement to do something in exchange for money. (EN: And how does this differ from "employment" of any other kind?)

The author speaks of mathematical approaches to determining the value of intangibles - in the mathematical approach to game theory, these are called "util"s - such that in the opening example of the guest choosing the smaller slice of cake, the person might have seen the act of choosing the larger piece to be worth three units, but the smaller slice to be worth five units (2 credits for the cake, three for preserving their self esteem).

However, he's not keen on this approach - while it helps to appreciate the full range of values that weigh on an individual's assessment, attaching numerals to sentiments is an arbitrary process that is prone to inaccuracy and oversimplification.

The Cake-Cutting Problem

The "cake-cutting problem" addresses the difficulty of finding an equitable means of dividng a finite resource in a manner that is acceptable to all parties.

The principle is "equal division of the contested sum: in any dispute, there is generally agreement to a point, so it's best to dole out the accepted measures first, then divide the remainder, in which there is argument, equally among the parties in dispute.

The example given is two men who pool their funds to buy a collection of books. Each man is permitted to take the books he wants but the other person does not. Then, the books that both parties want are divided equally. The result is that one party may get more than the other (one man may have 80 books and the other only 60), but each party got everything that was uncontested, and a fair share of the items that were in dispute.

It's suggested that this is appropriate to territorial disputes among political entities: where two factions claim overlapping borders in a disputed region, there is no need to consider the land they "own" that is not contested by the other party - only the area of overlap is to be contested.

(EN: My sense is that this works in an acceptable manner, provided that the parties involved fairly represent their claims. If a person is seeking to maximize their profit, they would claim far more than they really want, knowing that the outcome will give them a greater quantity than if they had presented an accurate smaller claim.)

While this works in instances where the item(s) in dispute are considered to be of equal unitary value (a book is a book, an acre of land is an acre of land), it is not so neat a problem when the units are of unequal value.

Returning to the cake example, the author mentions an instance in which a wedding cake is divided among guests. While the slices are more or less equal in size, the slices from the edge have more icing. The natural solution would be to dissect the cake, separating cake from icing, meting out the portions of each, and reassembling the portions to contain equal amounts of all three. (EN: the author suggests this as the best possible approach, but honestly, it sounds totally impractical - takes a lot of effort, and makes one heck of a mess.)

But even this "democratic" approach to determining portions is lacking, in that it calls for the situation to be managed by a central authority who arbitrarily dictates the terms of what individuals will consider equal. In the previous example, there are some people who really don't care for icing and would have been happier to have a slice with less, who are being compelled in the name of fairness to take something they do not want (and someone else does).