Game Theory

Game Theory Parables for Game Designers

Evolutionary Game Theory

Game Theory is typically thought of as a logical or rational or otherwise intellectual endeavour, akin to a mathematical analysis that only a rational agent (like a human) could do when looking at a game.  However, it is surprising to note that the lessons of Game Theory can be applied by entities that ostensibly aren’t rational, like lizards, squids, bacteria, and groups of Magic: the Gathering players.

These entities don’t go through the sorts of analysis that a rational human would do in order to find a Nash Equilibrium, for example, but the results of this analysis still apply to them.  The key is to view them as a population of players, interacting through games, and then passing on their strategies to the next generation (whether it be by reproduction or imitation) based on the outcome of the game.

The language of Game Theory can be translated into Evolutionary language as follows:

Player = replicator, which is anything that replicates itself and determines behavior

Game = an interaction between replicators that determines an outcome for all participants

Strategy = a behavioural trait that determines how the replicator behaves in the interaction

Utility = fitness, the average number of offspring that carry the behavioural trait into the next generation as a result of the trait being used in the current generation

Expected Utility = expected fitness

In this way, interactions between members of a population can be represented as a simultaneous move game, even though the “players” may not even be aware of the structure of the game they are playing.

Here’s a simple example: imagine a population of a species of bird where each bird can either be Aggressive or Passive.  This trait determines how they interact with other birds when they come across a resource, like a supply of food for example.

If an Aggressive bird meets a Passive bird, the Passive bird will surrender and the Aggressive bird will get all of the food.

If two Passive birds meet, they will share the food equally.

If two Aggressive birds meet, they will fight over the food.  Assuming that they are equally matched, each has a 50% chance of winning the food, while the loser incurs a cost for losing the fight.

Depending on the values of the food and the cost of losing the fight (in terms of reproductive fitness), this game can take on the form of a classic game like the Prisoner’s Dilemma, or Chicken.

Here’s an example where the value of the food resource is 3 units of fitness, and the cost of fighting is 5 units:

Chicken.PNG

This game has two Nash Equilibrium in pure solutions, but those aren’t possible in an evolutionary context where all players come from the same population (one Equilibrium, for example, says that one player always plays Passive, and the other always plays Aggressive; this requires the population to be both Aggressive and Passive at the same time, which doesn’t work).

There is a Mixed Strategy Nash Equilibrium, though, occurring when 60% of the population is Aggressive, and 40% is Passive.  This equilibrium describes a stable makeup of the population, a polymorphic equilibrium.

If the population is too far towards the Aggressive side (>60% Aggressive), then it is better to be born as a Passive bird, since the Aggressive birds will end up meeting each other, and fighting, too often.

If the population is too far towards the Passive side (>40% Passive), then it is better to be born as an Aggressive bird as there are many Passive birds to exploit.

What is interesting is that no strategy is clearly “better.”  They each play their role, and the true balance occurs when both are properly represented.  This often appears in highly interactive, economic games like Brass or Container, where the game really requires the right mix of different strategies. 

Game Theory’s insight is that this is a common, indeed expected, situation in an interactive game.  Designers should expect, and can encourage via their scoring systems, a stable “metagame” to emerge within any suitably interactive game.

This stability happens in nature, too.  The Common Side-Blotched Lizard (Uta stansburiana) is known as the Rock-Paper-Scissors Lizard for the variations that appear naturally in its population.

Orange-necked males are aggressive, and keep large harems of females.

Blue-necked males are less aggressive, and keep smaller harems. 

Yellow-necked males are sneaky, and look like females.

The yellow males can sneak into the orange males’ territory and steal one of their females (because the orange male’s territory is too large to defend), giving yellow an advantage over orange.  A blue male, however, has a smaller territory, and can more easily detect a yellow male interloper, giving blue an advantage over yellow.  A blue male, however, is not as aggressive as an orange male, and will lose territory in a fight, giving orange an advantage over blue.

This results in a pattern of cyclical dominance, as the population rotates through dominant colours.  With no external influence, it would stabilize at a point where each colour comprises 1/3 of the population.

Something like this could apply to the population of players who play a given game (as in a competitive CCG, for example).  Players, like lizards, will eventually settle into a polymorphic equilibrium driven solely from the internal rewards of the game (i.e. winning and losing).  Depending on the number of strategies available, and the payoffs of each interaction, highly varied metagames could emerge.  They could even include a strategy that, on the surface, looks weak, but has just the right interaction with its competitors to secure itself a small percentage of the metagame.  Like gears in a clock, the whole cannot function without the parts, no matter how small.

Sam Hillier