These assumptions are not fool-proof, however, since there are many arguments that could be made against them. Assumption 1: All the players act and think rationally , making choices and performing actions that are in their self-interest. The weakness of this assumption lies on the fact that human nature does not often permit rational thinking, and humans tend to make decisions for reasons other than their own interests. Assumption 2: All the players act strategically , while taking into consideration the responses of their competitors to their actions.
In fact, many business managers even do not make decisions within the strategic context. Assumption 3: The effectiveness of Game Theory is fully realized when all the players are fully aware of, and understand clearly, the payoffs — both the negative and positive ones — of their choices or actions.
It is a fact that most players make decisions even without having all the information. Many even do not care to know the full story behind some actions or responses before deciding on a specific course of action themselves. Game Theory can also be performed in two ways: simultaneous games , where the players make their moves or actions at the same time, without waiting for information on what the other players have chosen or acted upon, and sequential games , where the moves of the players will depend on, and in response to, the previous action or choice of another player.
Anyone that plays a key role in high-level and strategic decision-making in an organization should definitely consider learning about, and applying, the Game Theory. They include, but are not limited to, executives, directors, and senior managers. The applicability of Game Theory may be in question by some pundits, but there is no denying the fact that even large and established businesses have openly discussed using Game Theory for their key strategic business decisions.
Making business decisions is a daily event for managers. They are always faced with decisions on what to produce, what to procure, and what to sell, followed by decisions on how much they should spend in producing or in procuring, and what price they should set when they sell. There are so many tools used to come up with decisions, and one of these tools is Game Theory.
We can enumerate several reasons why business managers should consider using the Game Theory in its business operations. Business managers must beware, however, of how they use Game Theory. It is not a management tool that they can use as a substitute for experience in business. It is merely a tool, or a guide, for them to go about their tasks or roles as business managers.
The pricing decisions of a company can be highly influenced by the pricing choices or decisions of rival companies.
One popular example was the price-chopping decisions initiated by Intel and Advanced Micro Devices AMD on their desktop and mobile processors. Intel and AMD are considered to be competitors in a highly specialized niche, and both are in a tight race to gain a larger share of the market.
The first move was taken by Intel, who initiated a price slash on its desktop and mobile processors. AMD reacted by implementing a similar price cut, even if it meant potential losses or decrease in revenues. This price war resulted to both companies seeing significant increases in unit sales and shipments of their products — a sign of an increase in their market potential.
However, their revenues saw a drop, and so did the profits. The interrogators are not in possession of sufficient evidence or information to make a conviction, so they have to find a way to facilitate a confession.
They present each of the suspects with two choices: either defect , or confess to the crime and get a lighter punishment or jail sentence, or they could cooperate , or refuse to say anything, and suffer the punishment due them.
The interrogators make it a point to inform both suspects that the other is fully aware of the deal and its connotations. If the two suspects decide to defect and make a confession, they will both get the full brunt of the punishment and be sentenced the standard jailtime.
If, however, one suspect confesses and the other stays quiet, it is the latter who will get the jail sentence — often even longer — while the one who confessed gets off with a lighter sentence, or even walks away scot-free. Both companies are aware of each other as long-time competitors, and that they will be playing the same game for a long time to come.
Thus, they have the choice on whether to cooperate with each other and kept their prices higher, or they could engage in mutual price-chopping actions. Many assume that Game Theory applies only to getting the pulse of competitors.
However, it can also apply to the relationship of companies with their supply chain partners. Supply chain management can be facilitated by Game Theory concepts.
The game can be played in one of two ways. A non-cooperative game is where the players are unable to make binding commitments before making a choice or deciding what strategy to implement. On the other hand, making such commitments is possible in a cooperative game , which means that players can form coalitions and make side-payments. It helps validate a product concept early in your product development process.
It also helps product managers get user feedback as fast as possible to iterate and make small, incremental improvements to the product. Startups release the MVP to early customers then run experiments to gauge interest, test price sensitivity and messaging, and more.
It begins the feedback process to bring ideas and suggestions based on customer needs. This allows you to create iterations of the product and build something more valuable for your target market. While this phase will mainly involve finding manufacturers or suppliers , you may also factor storage, shipping, and warehousing into your choice.
In Shoe Dog , a memoir by Nike founder Phil Knight, the importance of diversifying your supply chain is a theme emphasized throughout the story. Finding multiple suppliers for the different materials you will need, as well as different potential manufacturers, will allow you to compare costs. Sourcing several options is an important part of safeguarding your business for the long term.
When looking for suppliers, there are plenty of resources both online and in person. While it may seem old fashioned, many business owners choose to attend trade shows dedicated to sourcing. Trade shows like Magic in Las Vegas provide the opportunity to meet hundreds of vendors at once—to see, touch, and discuss materials and build a personal relationship with suppliers, which can be valuable when it comes time to negotiate prices.
During the sourcing phase, you will inevitably come across the decision of whether to produce your product locally or overseas. It is a good idea to compare the two options, as they each have their own advantages and disadvantages. The most commonly used sourcing platform for overseas production is Alibaba. Alibaba is a marketplace for Chinese suppliers and factories, where you can browse listings for finished goods or raw materials.
A popular way of using Alibaba to find a manufacturer is to look for listings with similar products to your own and then contact the factory to see if they can produce your specific design.
After research, planning, prototyping, and sourcing is done, you should have a clearer picture of what it will cost to produce your product. Costing is a business analysis process where you take all information gathered thus far and add up what your cost of goods sold COGS will be so you can determine a retail price and gross margin.
Begin by creating a spreadsheet with each additional cost broken out as a separate line item. This should include all of your raw materials, factory setup costs, manufacturing costs, and shipping costs.
If you were able to secure multiple quotes for different materials or manufacturers during the sourcing phase, you can include different columns for each line item that compare the cost. Another option is to create a second version of the spreadsheet, so you can compare local production versus overseas production. Once you have your total COGS calculated, you can come up with a pricing strategy for your product and subtract the COGS from that price to get your potential gross margin, or profit, on each unit sold.
The last step in this methodology is to introduce your product to the market! At this point, a product development team will hand the reins over to marketing for a product launch. You can still run a successful go-to-market strategy by using the following tactics:. Struggling to grow sales? Learn how to go from first day to first sale in this free training course. These three industries have relatively straightforward paths to product development, thanks to the many well-documented case studies that can be used for inspiration.
In the fashion industry, product development usually begins the old-school way: with a hand-drawn sketch or the digital equivalent made using a program like Procreate. A sketch is then developed into a sample using a pattern maker or seamstress. During the prototyping phase, a size set is created, which means a range of samples with different measurements for each size you want to sell.
Once the size set is finalized, it is put into production. Rather than make the product, some fashion and apparel businesses choose print on demand to produce their clothing in the beginning.
Print-on-demand allows you to upload designs to a third-party app that connects your store with a warehouse and screen-printing facility. When an order is placed online, your design is printed on an existing stock of t-shirts, sweaters, or various other items on offer, creating a finished product without the need to design the entire garment. The beauty and cosmetics industry includes a wide range of products that is constantly expanding, due to wellness and self-care trends.
From makeup to bath products to skincare, many beauty brands are focusing on all-natural ingredients and sustainability, which makes it easier to prototype a product on your own using everyday ingredients.
White labeling is also popular in the beauty and cosmetics industry. Whichever route you decide to take, mass manufacturing for cosmetics is usually done by working with a lab and a chemist to make sure quality stays consistent at scale. Food and beverage products are among the easiest to start developing at a low cost and from the comfort of your own home.
In order to move from recipe to packaged goods you can sell in stores or online, you will need to find a commercial kitchen that is licensed to produce food and has passed a health and safety inspection.
These kitchens are usually set up with ovens and cooking equipment to accommodate large batches, but if you are considering mass production and packaging, a co-packer or co-manufacturer might be a better option. These are manufacturing facilities that specialize in processing raw materials and producing food and beverage products at scale.
Feeling uninspired? Watch some of the world's most successful entrepreneurs share their best advice for new business owners. One tradition in moral philosophy, utilitarianism, is based on the idea that all justifiable actions must be justified in this way. Without game theory, both of these problems resist analysis wherever non-parametric aspects are relevant. In doing this, we will need to introduce, define and illustrate the basic elements and techniques of game theory.
An economic agent is, by definition, an entity with preferences. Game theorists, like economists and philosophers studying rational decision-making, describe these by means of an abstract concept called utility. This refers to some ranking, on some specified scale, of the subjective welfare or change in subjective welfare that an agent derives from an object or an event.
For example, we might evaluate the relative welfare of countries which we might model as agents for some purposes by reference to their per capita incomes, and we might evaluate the relative welfare of an animal, in the context of predicting and explaining its behavioral dispositions, by reference to its expected evolutionary fitness.
In the case of people, it is most typical in economics and applications of game theory to evaluate their relative welfare by reference to their own implicit or explicit judgments of it. This is why we referred above to subjective welfare. Consider a person who adores the taste of pickles but dislikes onions. She might be said to associate higher utility with states of the world in which, all else being equal, she consumes more pickles and fewer onions than with states in which she consumes more onions and fewer pickles.
However, economists in the early 20th century recognized increasingly clearly that their main interest was in the market property of decreasing marginal demand, regardless of whether that was produced by satiated individual consumers or by some other factors. In the s this motivation of economists fit comfortably with the dominance of behaviourism and radical empiricism in psychology and in the philosophy of science respectively.
Like other tautologies occurring in the foundations of scientific theories, this interlocking recursive system of definitions is useful not in itself, but because it helps to fix our contexts of inquiry. When such theorists say that agents act so as to maximize their utility, they want this to be part of the definition of what it is to be an agent, not an empirical claim about possible inner states and motivations. Economists and others who interpret game theory in terms of RPT should not think of game theory as in any way an empirical account of the motivations of some flesh-and-blood actors such as actual people.
Rather, they should regard game theory as part of the body of mathematics that is used to model those entities which might or might not literally exist who consistently select elements from mutually exclusive action sets, resulting in patterns of choices, which, allowing for some stochasticity and noise, can be statistically modeled as maximization of utility functions.
On this interpretation, game theory could not be refuted by any empirical observations, since it is not an empirical theory in the first place. Of course, observation and experience could lead someone favoring this interpretation to conclude that game theory is of little help in describing actual human behavior.
Some other theorists understand the point of game theory differently. They view game theory as providing an explanatory account of actual human strategic reasoning processes. These two very general ways of thinking about the possible uses of game theory are compatible with the tautological interpretation of utility maximization. The philosophical difference is not idle from the perspective of the working game theorist, however. As we will see in a later section, those who hope to use game theory to explain strategic reasoning , as opposed to merely strategic behavior , face some special philosophical and practical problems.
Since game theory is a technology for formal modeling, we must have a device for thinking of utility maximization in mathematical terms. Such a device is called a utility function.
We will introduce the general idea of a utility function through the special case of an ordinal utility function. Later, we will encounter utility functions that incorporate more information. Suppose that agent x prefers bundle a to bundle b and bundle b to bundle c. We then map these onto a list of numbers, where the function maps the highest-ranked bundle onto the largest number in the list, the second-highest-ranked bundle onto the next-largest number in the list, and so on, thus:.
The only property mapped by this function is order. The magnitudes of the numbers are irrelevant; that is, it must not be inferred that x gets 3 times as much utility from bundle a as she gets from bundle c. Thus we could represent exactly the same utility function as that above by. The numbers featuring in an ordinal utility function are thus not measuring any quantity of anything. For the moment, however, we will need only ordinal functions.
All situations in which at least one agent can only act to maximize his utility through anticipating either consciously, or just implicitly in his behavior the responses to his actions by one or more other agents is called a game. Agents involved in games are referred to as players. If all agents have optimal actions regardless of what the others do, as in purely parametric situations or conditions of monopoly or perfect competition see Section 1 above we can model this without appeal to game theory; otherwise, we need it.
In literature critical of economics in general, or of the importation of game theory into humanistic disciplines, this kind of rhetoric has increasingly become a magnet for attack. The reader should note that these two uses of one word within the same discipline are technically unconnected. Furthermore, original RPT has been specified over the years by several different sets of axioms for different modeling purposes.
Once we decide to treat rationality as a technical concept, each time we adjust the axioms we effectively modify the concept. Consequently, in any discussion involving economists and philosophers together, we can find ourselves in a situation where different participants use the same word to refer to something different.
For readers new to economics, game theory, decision theory and the philosophy of action, this situation naturally presents a challenge. We might summarize the intuition behind all this as follows: an entity is usefully modeled as an economically rational agent to the extent that it has alternatives, and chooses from amongst these in a way that is motivated, at least more often than not, by what seems best for its purposes.
Economic rationality might in some cases be satisfied by internal computations performed by an agent, and she might or might not be aware of computing or having computed its conditions and implications. In other cases, economic rationality might simply be embodied in behavioral dispositions built by natural, cultural or market selection.
Each player in a game faces a choice among two or more possible strategies. The significance of the italicized phrase here will become clear when we take up some sample games below. A crucial aspect of the specification of a game involves the information that players have when they choose strategies. A board-game of sequential moves in which both players watch all the action and know the rules in common , such as chess, is an instance of such a game.
By contrast, the example of the bridge-crossing game from Section 1 above illustrates a game of imperfect information , since the fugitive must choose a bridge to cross without knowing the bridge at which the pursuer has chosen to wait, and the pursuer similarly makes her decision in ignorance of the choices of her quarry. The difference between games of perfect and of imperfect information is related to though certainly not identical with!
Let us begin by distinguishing between sequential-move and simultaneous-move games in terms of information. It is natural, as a first approximation, to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time. For example, if two competing businesses are both planning marketing campaigns, one might commit to its strategy months before the other does; but if neither knows what the other has committed to or will commit to when they make their decisions, this is a simultaneous-move game.
Chess, by contrast, is normally played as a sequential-move game: you see what your opponent has done before choosing your own next action. Chess can be turned into a simultaneous-move game if the players each call moves on a common board while isolated from one another; but this is a very different game from conventional chess. It was said above that the distinction between sequential-move and simultaneous-move games is not identical to the distinction between perfect-information and imperfect-information games.
Explaining why this is so is a good way of establishing full understanding of both sets of concepts. As simultaneous-move games were characterized in the previous paragraph, it must be true that all simultaneous-move games are games of imperfect information. However, some games may contain mixes of sequential and simultaneous moves. For example, two firms might commit to their marketing strategies independently and in secrecy from one another, but thereafter engage in pricing competition in full view of one another.
If the optimal marketing strategies were partially or wholly dependent on what was expected to happen in the subsequent pricing game, then the two stages would need to be analyzed as a single game, in which a stage of sequential play followed a stage of simultaneous play. Whole games that involve mixed stages of this sort are games of imperfect information, however temporally staged they might be. Games of perfect information as the name implies denote cases where no moves are simultaneous and where no player ever forgets what has gone before.
As previously noted, games of perfect information are the logically simplest sorts of games. This is so because in such games as long as the games are finite, that is, terminate after a known number of actions players and analysts can use a straightforward procedure for predicting outcomes.
A player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her.
She then asks herself which of the available final outcomes brings her the highest utility, and chooses the action that starts the chain leading to this outcome. This process is called backward induction because the reasoning works backwards from eventual outcomes to present choice problems. There will be much more to be said about backward induction and its properties in a later section when we come to discuss equilibrium and equilibrium selection.
For now, it has been described just so we can use it to introduce one of the two types of mathematical objects used to represent games: game trees. A game tree is an example of what mathematicians call a directed graph. That is, it is a set of connected nodes in which the overall graph has a direction. We can draw trees from the top of the page to the bottom, or from left to right. In the first case, nodes at the top of the page are interpreted as coming earlier in the sequence of actions.
In the case of a tree drawn from left to right, leftward nodes are prior in the sequence to rightward ones. An unlabelled tree has a structure of the following sort:. The point of representing games using trees can best be grasped by visualizing the use of them in supporting backward-induction reasoning. Just imagine the player or analyst beginning at the end of the tree, where outcomes are displayed, and then working backwards from these, looking for sets of strategies that describe paths leading to them.
We will present some examples of this interactive path selection, and detailed techniques for reasoning through these examples, after we have described a situation we can use a tree to model. Trees are used to represent sequential games, because they show the order in which actions are taken by the players. However, games are sometimes represented on matrices rather than trees. This is the second type of mathematical object used to represent games.
For example, it makes sense to display the river-crossing game from Section 1 on a matrix, since in that game both the fugitive and the hunter have just one move each, and each chooses their move in ignorance of what the other has decided to do.
Here, then, is part of the matrix:. Thus, for example, the upper left-hand corner above shows that when the fugitive crosses at the safe bridge and the hunter is waiting there, the fugitive gets a payoff of 0 and the hunter gets a payoff of 1. Whenever the hunter waits at the bridge chosen by the fugitive, the fugitive is shot.
These outcomes all deliver the payoff vector 0, 1. You can find them descending diagonally across the matrix above from the upper left-hand corner. Whenever the fugitive chooses the safe bridge but the hunter waits at another, the fugitive gets safely across, yielding the payoff vector 1, 0.
These two outcomes are shown in the second two cells of the top row. All of the other cells are marked, for now , with question marks. The problem here is that if the fugitive crosses at either the rocky bridge or the cobra bridge, he introduces parametric factors into the game. In these cases, he takes on some risk of getting killed, and so producing the payoff vector 0, 1 , that is independent of anything the hunter does. In general, a strategic-form game could represent any one of several extensive-form games, so a strategic-form game is best thought of as being a set of extensive-form games.
Where order of play is relevant, the extensive form must be specified or your conclusions will be unreliable. The distinctions described above are difficult to fully grasp if all one has to go on are abstract descriptions. Suppose that the police have arrested two people whom they know have committed an armed robbery together. Unfortunately, they lack enough admissible evidence to get a jury to convict.
They do , however, have enough evidence to send each prisoner away for two years for theft of the getaway car. We can represent the problem faced by both of them on a single matrix that captures the way in which their separate choices interact; this is the strategic form of their game:. Each cell of the matrix gives the payoffs to both players for each combination of actions.
So, if both players confess then they each get a payoff of 2 5 years in prison each. This appears in the upper-left cell. If neither of them confess, they each get a payoff of 3 2 years in prison each. This appears as the lower-right cell. This appears in the upper-right cell. The reverse situation, in which Player II confesses and Player I refuses, appears in the lower-left cell.
Each player evaluates his or her two possible actions here by comparing their personal payoffs in each column, since this shows you which of their actions is preferable, just to themselves, for each possible action by their partner.
Player II, meanwhile, evaluates her actions by comparing her payoffs down each row, and she comes to exactly the same conclusion that Player I does.
Wherever one action for a player is superior to her other actions for each possible action by the opponent, we say that the first action strictly dominates the second one. In the PD, then, confessing strictly dominates refusing for both players. Both players know this about each other, thus entirely eliminating any temptation to depart from the strictly dominated path.
Thus both players will confess, and both will go to prison for 5 years. The players, and analysts, can predict this outcome using a mechanical procedure, known as iterated elimination of strictly dominated strategies. Player 1 can see by examining the matrix that his payoffs in each cell of the top row are higher than his payoffs in each corresponding cell of the bottom row. Therefore, it can never be utility-maximizing for him to play his bottom-row strategy, viz.
Now it is obvious that Player II will not refuse to confess, since her payoff from confessing in the two cells that remain is higher than her payoff from refusing. So, once again, we can delete the one-cell column on the right from the game.
We now have only one cell remaining, that corresponding to the outcome brought about by mutual confession. Since the reasoning that led us to delete all other possible outcomes depended at each step only on the premise that both players are economically rational — that is, will choose strategies that lead to higher payoffs over strategies that lead to lower ones—there are strong grounds for viewing joint confession as the solution to the game, the outcome on which its play must converge to the extent that economic rationality correctly models the behavior of the players.
Had we begun by deleting the right-hand column and then deleted the bottom row, we would have arrived at the same solution. One of these respects is that all its rows and columns are either strictly dominated or strictly dominant. In any strategic-form game where this is true, iterated elimination of strictly dominated strategies is guaranteed to yield a unique solution. Later, however, we will see that for many games this condition does not apply, and then our analytic task is less straightforward.
The reader will probably have noticed something disturbing about the outcome of the PD. This is the most important fact about the PD, and its significance for game theory is quite general. For now, however, let us stay with our use of this particular game to illustrate the difference between strategic and extensive forms. In fact, however, this intuition is misleading and its conclusion is false. If Player I is convinced that his partner will stick to the bargain then he can seize the opportunity to go scot-free by confessing.
Of course, he realizes that the same temptation will occur to Player II; but in that case he again wants to make sure he confesses, as this is his only means of avoiding his worst outcome. But now suppose that the prisoners do not move simultaneously.
This is the sort of situation that people who think non-communication important must have in mind. Now Player II will be able to see that Player I has remained steadfast when it comes to her choice, and she need not be concerned about being suckered. This gives us our opportunity to introduce game-trees and the method of analysis appropriate to them.
First, however, here are definitions of some concepts that will be helpful in analyzing game-trees:. Terminal node : any node which, if reached, ends the game. Each terminal node corresponds to an outcome.
Strategy : a program instructing a player which action to take at every node in the tree where she could possibly be called on to make a choice. These quick definitions may not mean very much to you until you follow them being put to use in our analyses of trees below.
It will probably be best if you scroll back and forth between them and the examples as we work through them. Player I is to commit to refusal first, after which Player II will reciprocate when the police ask for her choice. Each node is numbered 1, 2, 3, … , from top to bottom, for ease of reference in discussion.
Here, then, is the tree:. Look first at each of the terminal nodes those along the bottom. These represent possible outcomes. Each of the structures descending from the nodes 1, 2 and 3 respectively is a subgame. If the subgame descending from node 3 is played, then Player II will face a choice between a payoff of 4 and a payoff of 3. Consult the second number, representing her payoff, in each set at a terminal node descending from node 3.
II earns her higher payoff by playing D. We may therefore replace the entire subgame with an assignment of the payoff 0,4 directly to node 3, since this is the outcome that will be realized if the game reaches that node.
Now consider the subgame descending from node 2. Here, II faces a choice between a payoff of 2 and one of 0. She obtains her higher payoff, 2, by playing D. We may therefore assign the payoff 2,2 directly to node 2. Now we move to the subgame descending from node 1.
This subgame is, of course, identical to the whole game; all games are subgames of themselves. Player I now faces a choice between outcomes 2,2 and 0,4. Consulting the first numbers in each of these sets, he sees that he gets his higher payoff—2—by playing D. D is, of course, the option of confessing. So Player I confesses, and then Player II also confesses, yielding the same outcome as in the strategic-form representation. What has happened here intuitively is that Player I realizes that if he plays C refuse to confess at node 1, then Player II will be able to maximize her utility by suckering him and playing D.
On the tree, this happens at node 3. This leaves Player I with a payoff of 0 ten years in prison , which he can avoid only by playing D to begin with. He therefore defects from the agreement. This will often not be true of other games, however.
As noted earlier in this section, sometimes we must represent simultaneous moves within games that are otherwise sequential. We represent such games using the device of information sets. Consider the following tree:. The oval drawn around nodes b and c indicates that they lie within a common information set. This means that at these nodes players cannot infer back up the path from whence they came; Player II does not know, in choosing her strategy, whether she is at b or c.
But you will recall from earlier in this section that this is just what defines two moves as simultaneous. We can thus see that the method of representing games as trees is entirely general.
If no node after the initial node is alone in an information set on its tree, so that the game has only one subgame itself , then the whole game is one of simultaneous play. If at least one node shares its information set with another, while others are alone, the game involves both simultaneous and sequential play, and so is still a game of imperfect information. Only if all information sets are inhabited by just one node do we have a game of perfect information.
Following the general practice in economics, game theorists refer to the solutions of games as equilibria. Note that, in both physical and economic systems, endogenously stable states might never be directly observed because the systems in question are never isolated from exogenous influences that move and destabilize them.
In both classical mechanics and in economics, equilibrium concepts are tools for analysis , not predictions of what we expect to observe. As we will see in later sections, it is possible to maintain this understanding of equilibria in the case of game theory. However, as we noted in Section 2.
For them, a solution to a game must be an outcome that a rational agent would predict using the mechanisms of rational computation alone. The interest of philosophers in game theory is more often motivated by this ambition than is that of the economist or other scientist.
A set of strategies is a NE just in case no player could improve her payoff, given the strategies of all other players in the game, by changing her strategy.
Notice how closely this idea is related to the idea of strict dominance: no strategy could be a NE strategy if it is strictly dominated. Now, almost all theorists agree that avoidance of strictly dominated strategies is a minimum requirement of economic rationality.
A player who knowingly chooses a strictly dominated strategy directly violates clause iii of the definition of economic agency as given in Section 2. This implies that if a game has an outcome that is a unique NE, as in the case of joint confession in the PD, that must be its unique solution.
We can specify one class of games in which NE is always not only necessary but sufficient as a solution concept. These are finite perfect-information games that are also zero-sum. A zero-sum game in the case of a game involving just two players is one in which one player can only be made better off by making the other player worse off.
Tic-tac-toe is a simple example of such a game: any move that brings one player closer to winning brings her opponent closer to losing, and vice-versa. In tic-tac-toe, this is a draw. However, most games do not have this property. For one thing, it is highly unlikely that theorists have yet discovered all of the possible problems. However, we can try to generalize the issues a bit.
First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically alert players would hit. Consider the strategic-form game below taken from Kreps , p. This game has two NE: s1-t1 and s2-t2.
Note that no rows or columns are strictly dominated here. But if Player I is playing s1 then Player II can do no better than t1, and vice-versa; and similarly for the s2-t2 pair. If NE is our only solution concept, then we shall be forced to say that either of these outcomes is equally persuasive as a solution. Note that this is not like the situation in the PD, where the socially superior situation is unachievable because it is not a NE. In the case of the game above, both players have every reason to try to converge on the NE in which they are better off.
Consider another example from Kreps , p. Here, no strategy strictly dominates another. So should not the players and the analyst delete the weakly dominated row s2?
When they do so, column t1 is then strictly dominated, and the NE s1-t2 is selected as the unique solution. However, as Kreps goes on to show using this example, the idea that weakly dominated strategies should be deleted just like strict ones has odd consequences.
Suppose we change the payoffs of the game just a bit, as follows:. Note that this game, again, does not replicate the logic of the PD. There, it makes sense to eliminate the most attractive outcome, joint refusal to confess, because both players have incentives to unilaterally deviate from it, so it is not an NE. This is not true of s2-t1 in the present game.
If the possibility of departures from reliable economic rationality is taken seriously, then we have an argument for eliminating weakly dominated strategies: Player I thereby insures herself against her worst outcome, s2-t2. Of course, she pays a cost for this insurance, reducing her expected payoff from 10 to 5.
On the other hand, we might imagine that the players could communicate before playing the game and agree to play correlated strategies so as to coordinate on s2-t1, thereby removing some, most or all of the uncertainty that encourages elimination of the weakly dominated row s1, and eliminating s1-t2 as a viable solution instead!
Any proposed principle for solving games that may have the effect of eliminating one or more NE from consideration as solutions is referred to as a refinement of NE. In the case just discussed, elimination of weakly dominated strategies is one possible refinement, since it refines away the NE s2-t1, and correlation is another, since it refines away the other NE, s1-t2, instead.
So which refinement is more appropriate as a solution concept? In principle, there seems to be no limit on the number of refinements that could be considered, since there may also be no limits on the set of philosophical intuitions about what principles a rational agent might or might not see fit to follow or to fear or hope that other players are following. We now digress briefly to make a point about terminology. This reflects the fact the revealed preference approaches equate choices with economically consistent actions, rather than being intended to refer to mental constructs.
Historically, there was a relationship of comfortable alignment, though not direct theoretical co-construction, between revealed preference in economics and the methodological and ontological behaviorism that dominated scientific psychology during the middle decades of the twentieth century. However, this usage is increasingly likely to cause confusion due to the more recent rise of behavioral game theory Camerer Applications also typically incorporate special assumptions about utility functions, also derived from experiments.
For example, players may be taken to be willing to make trade-offs between the magnitudes of their own payoffs and inequalities in the distribution of payoffs among the players. We will turn to some discussion of behavioral game theory in Section 8. For the moment, note that this use of game theory crucially rests on assumptions about psychological representations of value thought to be common among people. We mean by this the kind of game theory used by most economists who are not revisionist behavioral economists.
For a proposed new set of conventions to reduce this labeling chaos, see Ross , pp. Non-psychological game theorists tend to take a dim view of much of the refinement program. This is for the obvious reason that it relies on intuitions about which kinds of inferences people should find sensible. Like most scientists, non-psychological game theorists are suspicious of the force and basis of philosophical assumptions as guides to empirical and mathematical modeling.
Behavioral game theory, by contrast, can be understood as a refinement of game theory, though not necessarily of its solution concepts, in a different sense. It motivates this restriction by reference to inferences, along with preferences, that people do find natural , regardless of whether these seem rational , which they frequently do not. Non-psychological and behavioral game theory have in common that neither is intended to be normative—though both are often used to try to describe norms that prevail in groups of players, as well to explain why norms might persist in groups of players even when they appear to be less than fully rational to philosophical intuitions.
Let us therefore group non-psychological and behavioral game theorists together, just for purposes of contrast with normative game theorists, as descriptive game theorists. Descriptive game theorists are often inclined to doubt that the goal of seeking a general theory of rationality makes sense as a project. Institutions and evolutionary processes build many environments, and what counts as rational procedure in one environment may not be favoured in another.
On the other hand, an entity that does not at least stochastically i. To such entities game theory has no application in the first place.
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