Game Theory

Game Theory Parables for Game Designers


Here’s a game to consider.  Two people come across a delicious looking chocolate cake, just sitting there on a fancy place setting in the middle of the sidewalk.  Neither has any special claim to the cake, but they must decide how to divide it.

To avoid endless haggling, one player will make an offer to the other saying how much each person gets.  The other player then gets the choice to accept the offer, or decline it.  If they accept, then each player gets their share of the cake.  If they decline, however, no one gets anything.

If you were the proposer, what would you propose?

Most people, when presented with this question, give something close to a 50-50 split, maybe slightly biased towards themselves, but not noticeably so.  These splits are almost always accepted, as you would expect.

This raises a philosophical question: why do players prefer fairness?  I’ll come back to that shortly.

What happens in the case of a highly uneven split?  Say, 60-40 or even 70-30?  Again, most people tend to reject those offers, even though it means that they get nothing, instead of 30% or 40% of the cake. 

Why do players reject unfair offers when accepting them would mean they get something instead of nothing?  A simple answer is that the offer is unfair, but that just begets the question: what is so special about a fair outcome?

Traditional economic thought tends to disregard fairness, and say that the second player should accept any offer that gives them a non-zero amount of the cake.  Even an extreme 99-1 split should be accepted, this line of thought says, because 1% of a cake is better than 0% of a cake. 

The real interesting question is to ask why players reject unfair splits, and where this notion of fairness comes from.  Political philosophers have sought to answer this question for generations, but there is an answer from Evolutionary Game Theory that explains the evolution of fairness.

To set this up, the game needs to be simplified a bit.  Each player will submit their “bottom line,” the smallest amount of cake that they are willing to accept, written as a percentage.  Each player names their bottom line and these are revealed simultaneously.  If the two claims add up to less than 100%, then the players receive their claimed share.  However, if the two claims add up to more than 100%, no one gets anything.

Framed in this way, you can imagine a population of players playing this game with each other, and the cake they compete over determines their evolutionary fitness.  Each player in this population is defined by their bottom line, the minimum about of cake they are willing to accept.  If a “Demand 50%” player meets with another “Demand 50%” player, then they each get their share.  If, however, a “Demand 50%” player meets with a “Demand 60%” player, then no one gets anything as the total is over 100%.  Finally, there can also be inefficient cases: when a “Demand 40%” player meeting a “Demand 30%” player, they each get their share, but 30% of the cake goes to waste.

What sorts of strategies will emerge in this population?

An entire population that plays “Demand 50%” is in Nash Equilibrium.  If any player deviates to demand more, then they get nothing.  If they demand less, then they’ll simply get less.  Neither option is a better response to “Demand 50%,” so this truly is an Equilibrium.  This is great news for fairness, but it is not the end of the story.

Consider the interaction between “Demand 1/3” players and “Demand 2/3” players.  If they meet, each gets their share and nothing is wasted.  Neither player is better off by deviating from this pair, so this too is a Nash Equilibrium.  However, it is unfair: one player gets twice as much as the other.

What is striking is that a population where half of the people play “Demand 1/3” and the other half play “Demand 2/3” is a polymorphic equilibrium.  In other words, this unjust population is stable, just like the just and fair population of a 50-50 split.

In addition to being unjust and unfair, this population is also inefficient.  Whenever two “Demand 2/3” players meet, no one gets anything, and whenever two “Demand 1/3” players meet, the extra 1/3 of the cake is wasted.  The fair population does not have this problem of waste.

So the fair population, where everyone plays “Demand 50%,” is just one of many possible, stable, populations that could emerge.  How does a society like ours, where people play “Demand 50%” come to be?

Computer simulations in evolutionary game theory show the answer.  These simulations create a population of players playing randomized strategies, pair them up to play this game, and then use the outcome to determine which strategies pass into the next generation.  The process is repeated over hundreds of generations, and many different stable populations can emerge at the end.

Most, but certainly not all, of these simulations, end in an outcome where the whole population plays “Demand 50%.”  If the simulation is set up as above, only about 60% of the populations will end up at this fair society.  This means that the emergence of fairness certainly isn’t necessary, and it is barely more likely than not.

Certain changes to the simulations can make it more likely to occur, such as adding a tendency for players of the same strategy to play with each other.  I don’t want to go too deep down this simulation rabbit-hole, because there is more than enough to discuss right now.

This line of thought suggests that fairness emerged over time, through the course of many such interactions, and that embedded within us are our strategies/behavioural traits: we tend to prefer fair outcomes, we tend to reject unfair outcomes, and we tend to recognize and associate with other fair people (and shun those who are unfair).

Supporting this game theoretic analysis is some work in neuroscience.  People who are more empathetic tended to make more generous offers, suggesting that we have evolved an inner sentiment of fairness as part of our empathetic abilities.  Further, players who receive (and reject) unfair offers have significant brain activity in the region of the brain associated with disgust.  Again, this suggests that we have evolved an inner sentiment of disgust for injustice, for unjust outcomes and unfair people alike.

What does this all mean?  First, it is entirely natural for players to experience emotional responses to fair/unfair offers in the ultimatum game.  Second, this suggests why players have these reactions: we have evolved these emotive responses, these inner sentiments, due to the structure of the interactions we (and our ancestors) have participated in.  This evolutionary/historical process has forged people into the fair society that we have become.

(Third, and this is more philosophical than anything, it suggests the emergence of fairness, and the idea of fairness itself, is a purely natural phenomena.  It is not something that comes from reason, or from on high, or from any other heavy philosophical explanation).

What does this mean, finally, for game designers?  First, it is important to know that your players come pre-equipped with these notions of fairness, and that it is awfully tough to dislodge them from their inner sentiment of fairness by external means (by game points, for example).  Keep this in mind if your game features significant ultimatum-type interactions: you should expect players to reject unfair offers, even if that puts them in a worse-off position.

Second, if you know what your players are likely to do (and you know now that they are likely to prefer fairness), you can frame your dilemmas to take advantage of this.  Consider a mechanism that rewards unfair offers, or one that gives a player a different kind of incentive to accept an unfair offer.

Third, fair players like to associate with fair players, so give them a mechanism to do so.  Can you let players see who has done what in previous situations?  Can you give players a choice about who they interact with?  When it comes to players who favour unfair outcomes, can you give them an ability to signal their status as such?  This can also create opportunities for a push-your-luck game, a stare-down moment where players decide whether to be greedy, fair, or submissive.

The ultimatum game, players’ natural reactions to fairness and unfairness, and the conflict between the sentiment of fairness and the rational choice (of some cake over no cake) should provide fertile game design ideas.  A whole game can be built around the ultimatum game (as in NY Slice), but it could also be interestingly incorporated into a larger game.  Give players the opportunity to play the ultimatum game within a larger context and you can use these insights to inform that larger context.

Sam Hillier