Game: Split the Pie

Apr 17 2012

When a agreement between two parties bares fruit, there can be considerable contention on how to split the pie that is made from that fruit.  Some will try to base a proposed split on what is fair, others what is deserved.  And others still: what they can get away with.  Whether it’s a trade agreement, a government program, or even just purchasing a car, it so often seems that an agreement where all parties would benefit can break down from the inability to jointly decide how the dividends from the agreement should be divvied up.  Are there methods that can be employed to ensure that these opportunities are not foregone because of disagreements over how to share the pie?

The Game of Split the Pie

Let’s explore a simple Game Theory exercise that might help in considering possible tactics.  The game works like this:

There exists a $100 pot of money.  Two players secretly write down what amount of the $100 pot they wish to have in whole dollars.  The two amounts are then revealed and each player receives the amount they wrote plus 50% of the remainder of the pot.  However, if the sum of both amounts is greater than the $100 pot, both players receive nothing.

For example, if each player writes $50, then each player would receive $50 from the pot.  Similarly, if both players write $0, they both receive $50 since they would each be paid 50% of the remaining $100 in the pot.  However, if one player writes $80 and the other writes $40, the sum of the two amounts would be $120, and since there is not enough money in the $100 pot to cover both requests, they both receive $0.

How to Split the Pie

Splitting the pie is not the same as a zero sum game in which one player’s gain is the other player’s loss.  Although this kind of game can often be mistaken for zero sum, both players can receive a payment that makes them better off than before playing, even if it’s only a very small payment.  Yet this is also where the difficulty lies.  A player who assumes this idea would perhaps write $1 with the idea that doing so will result in the best chance of receiving anything from the pot.  Should they encounter another player with a similar tactic, they will both split the pot evenly and walk away with $50 each.  In fact, the pot is split evenly when both players write the same amount between $0 and $50.

But what if you know you will encounter a player who will ask for a dollar?  The temptation is to ask for $99.  And why shouldn’t you?  Both players will be better off than before, you will just happen to be relatively much better off than the other player.

The Problem

There is no real equilibrium to this game and therein lies the problem.  There is no right strategy.  While this is a simplified game in that there is only one round and no open negotiation, this problem arises, even in the form of this simple game, more often than we might think.  Consider a scenario where two parties are negotiating the price of a car.  The potential buyer may make an offer so low that the seller does not even consider further negotiation, even if the buyer might be willing to make a much higher offer, so the first round becomes the only round.  A trade that would benefit both parties cannot be completed because at least one party was relatively too greedy in their ask.

However, within the confines of the one round, secret ballot rules, each player is on their own.  My guess is that if played in an experimental setting, most players would choose $50.  Most would consider it fair and if the other player asks for more then it would “serve them right” for causing the breakdown of the deal and causing both players to receive a $0 payout.

What Can be Done?

Add a new rule to the game: no player may ask for more than $50.  In real life, this might take the form of a government regulator.  Of course, if you’re playing against people only asking for a dollar, you would lobby hard against this kind of regulation.

Or perhaps add a rule that arbitrates so that a deal can always be made.  This might be in the form of a penalty to the player whose ask amount is greater amount that would break the deal that would be paid to the other player.  Of course, if the goal is simply to ensure that deals are always made so that both players are always better off, then this arbiter could even be random in it’s “fairness”.

Concluding agreements that result in a pie, no matter how that pie is split, benefit everyone involved.  So what other tactics might be employed to ensure that these deals are successfully reached more often?




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