Elements of Games
A game in game theory is any situation involving the interdependence between adaptive agents as they make choices that affect each other and the overall outcome of the situation. Games can be identified as having a number of central elements, which can be identified as players, strategies, and payoffs.1 In order to have a game it is necessary to have some specification for each of these elements before the game can be modeled coherently.
The players in a game are called agents. Agents are abstract models of individuals or organizations which have agency. Agency means the capacity of actors to make choices and to act independently on those choices to affect the state of their environment; they do this in order to improve their state within that environment. In order to act and make choices, agents need a value system and need some set of rules under which to make their choices so as to improve their state with respect to their value system.2 Of central importance here is the idea of rationality. A dictionary definition of rationality would read something like this: based on or in accordance with reason or logic.3 Rationality means acting according to a consistent set of rules, that are based upon some value system that provides the reason for acting.
To act rationally is to have some value system and to act in accordance with that value system. When a for-profit business tries to sell more products it is acting in a rational fashion, because it is acting under a set of rules to generate more of what it values. When a person who values their community does community work they are acting rationally. Because their actions are in accordance with their value system and thus they have a reason for acting in that fashion. Standard game theory makes a number of quite strong assumptions about the agents involved in games. A central assumption of classical game theory is that the players act according to a limited form of rationality, what is sometimes call hyperrationality.4 A player is rational in this sense if it consistently acts to improve its payoff without the possibility of making mistakes, has full knowledge of other players’ intentions and the actions available to them, and has an infinite capacity to calculate a priori all possible refinements in an attempt to find the “best one”. If a game involves only rational agents, each of whom believes all other agents to be rational, then theoretical results offer accurate predictions of the games outcomes.5
Agents have a single conception of value, i.e. all value is reduced to a single homogeneous form called utility. Preferences and value are well-defined. Rational agents have unlimited rationality, the idea of omnipotence, i.e. they know all relevant information when making a choice, they can compute this information and all of its consequences. Within this model, agents have perfect information and any uncertainty can be reduced to some probability distribution. The agent’s behavior is then seen to be an optimization algorithm over their set of possibilities.6
Game theory is a young field of study—less than a century old. In that time, it has made remarkable advances, but it remains far from complete. Traditional game theory assumes that the players of games are rational—that they act in best accordance with their own desires given their knowledge and beliefs. This assumption does not always appear to be a reasonable one. In certain situations, the predictions of game theory and the observed behavior of real people differ dramatically.
People in the real world operate according to a multiplicity of motives, some of the time people are in a situation where they are simply trying to optimize a single metric, but more often they are not. They are embedded within a context where they are trying to optimize according to a number of different metrics.7
The fact that people are not always optimizing according to a single metric is illustrated in the many games where people do not choose actions that give them the greatest payoff within that single value system. The best empirical examples of this are taken from the dictator game. The dictator game is a very simple game, where one person is given a sum of money, say 100 dollars, this person plays the role of “the dictator,” and is then told that they must offer some amount of that money to the second participant, even if that amount is zero. Whatever amount the dictator offers to the second participant must be accepted. The second participant, should they find the amount unsatisfactory, cannot then punish the dictator in any way.
Standard economic theory assumes that all individuals act solely out of self-interest. Under this assumption, the predicted result of the dictator game is that the “dictator should keep 100% of the cake, and give nothing to the other player.” This effectively assigns the value of what the dictator shares with the second player to zero. The actual results of this game, however, differ sharply from the predicted results. With a “standard” dictator game setup, “only 40% of the experimental subjects playing the role of dictator keep the whole sum.” In research by Robert Forsythe and al. they found the average amount given, under these standard conditions, to be around 20% of the allocated money.8 In the majority of these game trials, the dictator assigns the second player a non-zero amount.
The obvious reason for this is that the dictator is not simply trying to optimize according to a single monetary value - that a strict conception of rationality would posit - but is acting rationally to optimize according to a number of different value systems. They want the money, yes, but they are also optimizing according to cultural and social capital that motivates them to act in accordance with some conception of fairness and it is out of the interaction of these different value systems that we get the empirical results.
What agents value can be simple or it can be complex. A financial algorithm is a form of agent that acts according to some set of rules designed to create a financial profit; this is an example of a very simple value system. In contrast what a human being values is typically many things. People value social capital, that is to say, their relationships with other people and their roles within social groups. They care about cultural capital how they perceive themselves and how others perceive them. They care about financial capital and natural capital. They often care about their natural environment to a greater or lesser extent. Likewise, the set of instructions or rules can be based on some simple linear cause and effect model - what may be called an algorithm - or they may be much more complex models - what may be called a schema.
Thus when we say that someone is acting rationally and maximizing their value payoff, this can be in many different contexts. A person helps an old lady onto the bus, not because they are going to get paid for this, but what they do get from this is some sense of being a decent person and they gain some payoff in that sense. Thus it is not the concept of rationality or that people try to optimize their payoff that needs to be revised. It is the narrow definition of rationality as optimizing according to a single metric that needs to be expanded within many contexts that involve social interaction.9
The classical conception of strict rationality based upon a single metric will apply in certain circumstances. It will be relevant to many games in ecology, where creatures have a simple conception of value maximization. Likewise, it will often be relevant to computer algorithms and software systems and sometimes relevant for socioeconomic interactions, or at least partially relevant. As the influential biologist Maynard Smith, in the preface to the book Evolution and the Theory of Games noted “paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behavior for which it was originally designed.”10 In order to achieve an empirically accurate theory of games between more complex agents it will likely need to be expansive in its conception of value and rationality to include the more complex set of values and reasoning processes that are engendered in such games.
Strategy is the choice of one’s actions. In game theory, player’s strategy is any of the options they can choose in a setting where the outcome depends on the action of others. A strategy, in the practical sense, is then, a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.11 For example, the game might be a business entering a new market and trying to gain market share against other players. This will not just happen overnight but they will have to take a series of actions that are all coordinated towards their desired end result. They might first have to organize production processes and logistics, then advertising, then pricing etc. Each of these actions we would call a move in the game, and the overall strategy consists of a set of moves.
A player’s strategy set defines what strategies are available for them to play. For instance, in a single game of rock-paper-scissors, each player has the finite strategy set of rock, paper, scissors. Likewise, a player’s strategy set can be infinite, for example in choosing how much to pay when making an offer to purchase an item in a process of bartering, this could be potentially infinite, it could be any increment.
Pure / Mixed Strategy
In some games, there will not be one primary strategy that an agent will always choose but in many circumstances, they may have a number of options and choose between them with some given probability. This will often be the case when they do not want the other player to know in advance which move they will take. For example, in smuggling goods across the Vietnam-Chinese border, the smugglers have many different points of entry available to them and the police have many different points that they could secure. In such a case neither side wants to always choose the same location they want some degree of randomness in the strategy that they choose.
This gives us a distinction in games between those with strategies that one will always play and those that one will play only with a given probability, this distinction is captured in the terms mixed and pure strategy. Pure strategies are ones which do not involve randomness and tell us what to do in every situation. A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation they may face. Strategies that are not pure—which depend on some aspect of chance—are called “mixed strategies.” In mixed strategies, you have a number of different options and you ascribe a probability to the likelihood of playing them. As such we can think about a mixed strategy as a probability distribution over the actions players have available to them.12
For every strategy taken within a game, there is a payoff associated with that strategy. A player’s payoff defines how much they like the outcome of the game. The payoffs for a particular player reflect what that player cares about, not what another player thinks they should care about. Payoffs must reflect the actual preferences of the players, not preferences anyone else ascribes to them. Game theorists often describe payoffs in terms of utility—the general happiness a player gets from a given outcome. Payoffs can represent any type of value, but only the factors that are incorporated into the model, thus we have to be careful in asking what do the agents really value. Payoffs are then essentially numbers which represent the motivations of players. In general, the payoffs for different players cannot be directly compared, because they are to a certain extent subjective.13
Payoffs may have numerical values associated with them or they may simply be a set of ranking preferences. If the payoff scale is only a ranking, the payoffs are called “ordinal payoffs.” For example, we might say that Kate likes apples more than oranges and oranges more than grapes. However if the scale measures how much a player prefers one option to another, the payoffs are called “cardinal payoffs.” So if the game was simply one for money then we could ascribe a value to each payoff, that would be the quantity of money gained.
In many games all that matters is the ordinal payoffs, all we need to know is which options they prefer without actually knowing how much they prefer them. This is useful because typically people do not ascribe specific values to how much they like things, but they do think about whether they prefer one thing or another. Kate may know that she likes apples more than oranges but she would probably laugh if you asked her to put values on how much more she likes them.