After stating these three facts, Euler concludes his proof with Paragraph 21, which simply states that after one figures out that a path exists, they still must go through the effort to write out a path that works. Euler believed the method to accomplish this was trivial, and he did not want to spend a great deal of time on it.
However, Euler did suggest concentrating on how to get from one landmass to the other, instead of concentrating on the specific bridges at first. He used capital letters to represent landmasses, and lowercase letters to represent bridges.
This was a completely new type of thinking for the time, and in his paper, Euler accidentally sparked a new branch of mathematics called graph theory, where a graph is simply a collection of vertices and edges.
Today a path in a graph, which contains each edge of the graph once and only once, is called an Eulerian path, because of this problem. From the time Euler solved this problem to today, graph theory has become an important branch of mathematics, which guides the basis of our thinking about networks. The origins of graph theory are humble, even frivolous … The problems which led to the development of graph theory were often little more than puzzles, designed to test the ingenuity rather than the stimulate the imagination.
But despite the apparent triviality of such puzzles, they captured the interest of mathematicians, with the result that graph theory has become a subject rich in theoretical results of a surprising variety and depth. As Biggs' statement would imply, this problem is so important that it is mentioned in the first chapter of every Graph Theory book that was perused in the library. Some other graph theory problems have gone unsolved for centuries [ ScienceWeek, 2].
This meant that only two landmasses had an odd number of links, which gave a rather straightforward solution to the problem. German civilians begin to evacuate from the town, but move too late. Thousands of people are killed trying to flee by boat and on foot across the icy waters of the Curonian Lagoon.
This map shows how much the town has changed. Many of the bridges were destroyed during the bombings, and the town can no longer ask the same intriguing question they were able to in the eighteenth century. Biggs, Norman L. Lloyd, and Robin J. Graph Theory: Oxford: Clarendon Press, Dunham, William.
Euler: The Master of Us All. Washington: The Mathematical Association of America, Hopkins, Brian, and Robin Wilson. Editor's note: This article was originally published in Convergence, Volume 3 Skip to main content.
Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M. Author s :. Euler and the Bridge Problem Why would Euler concern himself with a problem so unrelated to the field of mathematics? In a letter written the same year to Giovanni Marinoni, an Italian mathematician and engineer, Euler said [quoted in Hopkins, 2], This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it.
Euler's Proof On August 26, , Euler presents a paper containing the solution to the Konigsberg bridge problem. The probabilities are computed so as to maximize the expected point gain independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each player. This method can compute provably optimal strategies for all two-player zero-sum games. Player one will then win 2. Non Zero-Sum game The most famous example of a non-zero-sum game is the Prisoner's dilemma, as mentioned above.
Any gain by one player does not necessarily correspond with a loss by another player. The 'kill or be killed' business ideal are non zero-sum games. For example, a business contract ideally is a positive-sum game, where each side is better off than if they didn't have the contract.
Most games that people play for recreation are zero-sum. Cooperative games are those in which the players may freely communicate among themselves before making game decisions and may make bargains to influence those decisions. Monopoly can be a cooperative game, while the Prisoner's dilemma is not. However, Monopoly is a zero-sum game as there can be only one winner, whereas the Prisoner's dilemma is a non-zero-sum game. Most of life can be described as a cooperative game, because we normally cooperate against our opponents.
Complete information games are those in which each player has the same game-relevant information as every other player. Chess and the Prisoner's dilemma are complete-information games, while Poker is not. Not much of life can be described as complete information game. For the above example to work, the participants in the game have to be assumed to be risk neutral.
However, in reality people are often risk averse and prefer a more certain outcome - they will only take a risk if they expect to make money on average. Subjective expected utility theory explains how a measure of utility can be derived which will always satisfy the criterion of risk neutrality, and hence is suitable as a measure for the payoff in game theory.
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. Select a reason and click "Flag Post" to flag this for review. You may provide an optional required if choosing other description of why you find this objectionable. You may also wish to send a private message to to request him or her to edit or remove the. Email or username:. Log In Forgot your password? Results Leaderboard Upcoming Games. Join Bridge Winners Log in with:.
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