 Games, in game theory, involve a number of central elements which can be identified as players, strategies and payoffs. In this video we're going to zoom in to better understand each of these different elements to a game, talking firstly about the players and rationality, before going on to look at strategies and payoffs. As we've touched upon in a previous video, agents are abstract models of individuals or organizations which have agency. Agency is the capacity of an actor to make choices and to act independently on those choices and 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. A central idea here is that of rationality and we have to be careful how we define this idea of rationality. Addictionary definition of rationality would read something like this based on or in accordance with reason or logic. Rationality simply 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 also 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 the games. A central assumption of classical game theory is that players act according to a limited form of rationality, what is sometimes called hyper rationality. A player is considered rational in this sense if it consistently acts to improve its payoff without the possibility of making mistakes, has full knowledge of other players interactions and the actions available to them and has an infinite capacity to calculate beforehand all possible refinements in an attempt to find the best option. If a game involves only rational agents in this sense, each of whom believes all other agents to also be acting rationally then theoretical results from classical game theory offer accurate predictions of the games outcomes. In this interpretation of rationality agents have a single conception of value that is to say all value is reduced to a single homogeneous form often called utility. Preferences and value are well defined, rational agents have unlimited rationality which is the idea of omnipotence that is to say 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. In this model an agent's behavior is then seen to be an optimization algorithm over the set of possibilities available to them in the game. Game theory is a relatively young field of study as it's less than a century old. In this time we can say that it's made remarkable advances but the general theory remains far from complete. Traditional game theory assumes that the players of the game are hyper-rational but 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 actually happen to 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're not. They are embedded within a context where they're trying to optimize according to a number of different metrics. The fact that people aren't always optimizing according to a single metric is illustrated in the many games where people don't 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 a hundred dollars. This person plays the role of the dictator and is then told that they must offer some amount of money to the second participant even if that amount is zero. Whatever amount the dictator offers to the second participant must be accepted by that person. 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 the game however differ sharply from the predicted results. With a standard dictator game setup only 40% of the experimental subjects playing the role of the dictator keep the whole sum of money. In research conducted by Robert Forsythe and Al they found the average amount given under these conditions to be around 20% of the allocated money. In any case in the majority of these trials the dictator assigns the second player a non-zero sum amount in contrary to standard predictions. 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 is out of the interaction of these different value systems that we get the empirical results. What agent's 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 different 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 we might call an algorithm or there may be much more complex models what we might call 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're 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. That form of cultural capital cannot be reduced to a single monetary metric. Thus it is not the concept of rationality or that people are trying 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. The classical conception of strict rationality based on a single metric will apply in certain circumstances. It will be relevant to many games in ecology where creatures have a single conception of value maximization. Likewise it will often be relevant to computer algorithms and software systems and sometimes relevant to socio-economic interactions or at least partially relevant. As the influential biologist Maynard Smith noted in the preface to the book Evolution and the Theory of Games quote 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. If we want an empirically accurate theory of games between more complex agents it will need to be expansive in its conception of value and rationality to include the more complex set of value systems and reasoning processes that are engendered in such games. We've spent quite a bit of time talking about this idea of rationality as it is a major unresolved flaw within standard game theory and one that is important to be aware of. Strategy in the general sense 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 a game telling a player what to do for every possible situation throughout the game. 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 these individual moves. A player's strategy set defines what strategies are available for them to play. For example in a single game of rock paper scissors each player has the finite strategy set of rock paper or 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 bargaining this could potentially be infinite as it could be any increment. In some games there will 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 don't want other players to know in advance which move they will make. 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 along the border that they could potentially 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 some given probability. This distinction is captured in the term 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 every situation they face. Strategies that are not pure that depend on an element 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 any given strategy. As such we can think about a mixed strategy as a probability distribution over the actions players have available to them. 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 to the game. The payoffs for a particular player reflects what that player cares about and not what other players think that 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 as this is clearly foundational to the model. Payoffs are then essentially numbers which represent the motives of the players. In general the payoffs for different players cannot be directly compared because they are to a certain extent subjective. Payoffs may have a numerical value associated with them or there 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. This is an ordinal ranking. 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 describe a value to each payoff. There would be a quantity of money and this would be a cardinal payoff. In many games all that matters is the ordinal payoffs. All we need to know to construct our model is which options they prefer without actually knowing how much they prefer them. This is useful because in reality people don't go around ascribing specific values to how much they like things but they do think about whether they prefer one thing to another. Kate may know that she likes apples more than oranges but she would probably laugh if you asked her to put a value on how much more she likes them. So these are the three elements that are essential to modeling games players strategies and payoffs. In the next section to the course we'll start to play some actual games looking at how to solve games how to find the best strategies and talk about the important idea of equilibrium.