Game Theory

Game Theory Parables for Game Designers

Rational Player Behaviour

The value of Game Theory is how it allows you to understand Player behaviour:

  • What are they going to do?

  • What do I want them to do?

  • How do I stop them from doing this?

This could be useful after play-testing if you’re noticing players doing / not doing something you want them to. The tools of Game Theory can help you nudge players in a certain direction.

In order to be able to predict player behaviour, Game Theory needs to make a few assumptions about players. This is the source of much of the controversy around Game Theory, which often arises when the analysis or solution produced by Game Theory doesn’t match what players do in the real world (or, more commonly, what people think they should do, what seems right). Understanding what exactly Game Theory assumes about players can help resolve some of these controversies.

Game Theory assumes that players are Rational — this simply means that players act to maximize expected utility. Rationality doesn’t tell you what to want; it tells you how to get what you want. In this, it follows an idea from the philosopher David Hume:

“Reason is, and ought only to be the slave of the passions, and can never pretend to any other office than to serve and obey them”

David Hume, A Treatise of Human Nature (1739)

Hume is saying that our passions, our desires, are preferences, our feelings are primary. Reason, rationality, logic and the like cannot determine what we want, what we desire. Their role is only to guide us to these passions in the most effective way possible. This is the basis of rationality in Game Theory.

Game Theory assumes that players act to maximize expected utility, and Utility here is basically a numerical measure of preference. That is, the utility of one outcome is higher than another just in case the player prefers the one outcome over the other. Players' passions determine their utilities, and rationality helps them maximize those utilities.

Utility represents a player’s preferences, it does not determine them. Here’s yet another quote from David Hume that captures this idea:

“It appears evident that the ultimate ends of human actions can never, in any case, be accounted for by reason, but recommend themselves entirely to the sentiments and affections of mankind, without any dependence on the intellectual faculties”

David Hume, Enquiry Concerning the Principles of Morals (Appendix I)

It is common to see Game Theory talk in terms of financial outcomes (after all, its primary applications are in economics and politics) or other tangible outcomes like years in prison (as in the Prisoner’s Dilemma). This lends itself to a natural criticism of Game Theory, that it only cares about money, or selfish outcomes, but that is not true at all. What players value varies from player to player, and from game to game.

In a game of Agricola, a player may need 3 stone to finish a renovation, so at that point of the game, the utility of stone is higher than that of the pile of clay that is also available. A different player (or the same player at a different time of the game) may have different preferences — they may need the clay more than the stone, so it would be assigned a higher utility.

In a semi-cooperative game like Dead of Winter, one player may prefer that the colony succeeds even if they fail to meet their individual objective, while another may prefer that everyone loses since that player is unable to meet their objective. These players will have radically different utilities assigned to the very same outcomes.

Understanding these utility assignments will let you predict how those players are going to act. As a fellow player, you can then think strategically, and use these beliefs to change your own actions: act in ways that get the other players to act the way you need to. As a game designer, if you understand how players are going to act, you can better facilitate the actions you want and impede those you don’t.

Returning to the Prisoner’s Dilemma, there was a tacit assumption that players preferred to spend less time in prison than more, and that they placed no value on the time spent in prison by the other player. If this assumption isn’t true, then the analysis of the game is incorrect.

If players in the Prisoner’s Dilemma were friends, or they valued the good of the group over their individual fate, then the would assign different utility values to the outcomes (as opposed to a straight, one-to-one correlation between utility and years spent in prison). Critically, if the outcome where one player plays Betray while the other plays Quiet gets a different utility value (one that is lower than the utility from both players staying Quiet) then the game changes dramatically. Indeed, it stops being a Prisoner’s Dilemma and becomes a Stag Hunt, where both mutual Betrayal and mutual Cooperation are possible, stable outcomes.

This is the key lesson of the Prisoner’s Dilemma. In its original form, mutual cooperation for the greater good is impossible. To make it possible, you need to change the game. Of course, then a different problem emerges: how to get players to cooperate in the Stag Hunt. This game is at least solvable, though.

Returning to the discussion of Utility itself, the definition also means that it is not irrational for players to prefer whatever they wish. They may even assign lower utility to risky situations -- risk aversion -- or higher utility to risky situations -- risk seeking. Designers need to think carefully about what the players will want to do, how they will react to game situations, how that determines the utility they assign to various outcomes, and how this ultimately determines their actions and the strategic actions of all other players.

Sam Hillier