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Topic: Payoff matrix

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 Normal form game - Wikipedia, the free encyclopedia
A payoff function for a player is a mapping from that the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility - often cardinal in the normal form representation) of a player, i.e.
For example, in the game matrix on the left if a payoff is emboldened, it means that player whose payoff it is has a best response to the corresponding strategy played by the other player by playing that strategy.
In the matrix on the left, payoffs are emboldened as before to show the best response of the player whose payoff it is (note that a best response is the strategy according to that payoff, not the payoff).
en.wikipedia.org /wiki/Normal_form_game   (824 words)

 Facts about topic: (Payoff matrix)   (Site not responding. Last check: 2007-10-06)
A payoff matrix or payoff function is a concept in game theory ((economics) a theory of competition stated in terms of gains and losses among opposing players) which shows what payoff each player will receive at the outcome of the game.
Often these two payoff matrices are combined into a single matrix representation.
Note that the payoff matrices themselves are not symmetric (additional info and facts about symmetric) matrices however.
www.absoluteastronomy.com /encyclopedia/p/pa/payoff_matrix.htm   (855 words)

 [No title]
            Given any payoff matrix, the standard assumption is that the players choose their strategies simultaneously (in any event, that each player must choose a strategy in ignorance of the strategic choice of the other player — or, as described in the original handouts, by “secret ballot”) and without pre-play communication.
For simplicity, suppose the payoff matrix is 2 × 2, i.e., there are two just two players, each with a choice between two strategies, so the game has just four possible outcomes (cells in the matrix).
And suppose that the payoff matrix is common knowledge to each player, i.e., each player knows what the other player’s payoffs (interests/preferences/values are).
userpages.umbc.edu /~nmiller/POLI388/VARIATIONS.htm   (685 words)

 Minimax Theorem
Note that only the payoffs, presented as number of days of bombing, for the allied forces are shown.
This payoff matrix indicates that if the Japanese sail north and the allies search south, then the allies will spend one day determining that the Japanese did not sail south and then spend another day finding the Japanese ships to the north.
For (Sun,Up) the fighter payoff is 0.95 but the fighter has an incentive to attack from the bottom because the payoff increases to 1.
students.cs.byu.edu /~cs670ta/Lectures/Minimax2.html   (1784 words)

 [No title]
In the very simple payoff matrix games we examine in POLI 388, a (pure) strategy for the row player is simply a row in the matrix and likewise a pure strategy for the column player is simply a column in the matrix.
Remember that the payoff matrix for a zero-sum game may have just one number in each cell, representing the payoff for the row player P1.
The payoff for the column player P2 is simply the negative of the payoff for P1 (and therefore the sum of the two payoffs associated with each cell is always zero).
userpages.umbc.edu /~nmiller/POLI388/MIXED.Q&A.htm   (874 words)

Solution: In order to reduce the payoff matrix, we repeatedly look for rows and columns which are dominated by other rows (or columns).
Notice that for the row corresponding to strategy A, the first number in each column (the payoff for the row player) is less than the corresponding number for strategy D.
That is, strategy (D, B) is the equilibrium point, with a payoff of 6 for the row player, and 9 for the column player.
www.math.sunysb.edu /~scott/mat118.spr03/quizzes/q9-sol   (266 words)

 Matrix Operations
The result is a new matrix with the same dimensions in which each element is the sum of the corresponding elements of the previous matrices.
The inverse of a square matrix is a matrix of the same size that, when multiplied by the matrix, gives an identity matrix of the same size.
Matrix inversion is often used to solve a set of simultaneous linear equations.
www.stanford.edu /~wfsharpe/mia/mat/mia_mat2.htm   (1904 words)

 Innovation1 Introduction   (Site not responding. Last check: 2007-10-06)
The tendency of randomly generated payoff matrices (ecologies) to exhibit cyclic effects in the replicator dynamics.
The randomization of the frequencies is uniform over the simplex, so modulo the usual statistical uncertainties, the rate at which the system goes to an equilibrium from a randomized restart should be proportional to the sizes of the basins of attraction.
When any type's frequency goes to zero, the row and column for that type in the payoff matrix are replaced by new random numbers.
www.ethics.ubc.ca /eame/eameweb/Simulators/Simulators/innovation/innovati.htm   (396 words)

 Notes to Evolutionary Game Theory
Of course, Nowak and May were speaking somewhat loosely when they referred to this behavior as "chaotic." Since there are only finitely many states of the population, it must be the case that this dynamical system will eventually settle into a cycle (although it may not repeat itself for a very long time).
In both of the previous cases, it is easy to predict the future state of the population given its current state.
In the Hawk-Dove game discussed in section 2, the quantities in the payoff matrix represent the change in Darwinian fitness of the two individuals.

 The Beauty of Gray
Either by actually overthrowing their governments militarily or by the threat of doing so, the US needs to change their payoff matrix so that support of Palestinian terrorism is no longer in their interest.
Payoff matrices also are a very clean and terse way to present a lot of data, in a more or less (in this case less) quantitative way, which is how I like to think of things.
Further, for simplicity I’ll assume that the payoffs are symmetric—that the Israeli payoff if they fight and the Palestinians negotiate is the same as the payoff for the Palestinians if they fight and the Israelis negotiate.
beautyofgray.blogspot.com /2002_04_07_beautyofgray_archive.html   (3602 words)

 Minimax and Alpha-Beta Pruning
Payoffs for me (and losses for you) are listed in the middle cells.
Notice that the maximin and minimax values are symmetric; if I define the payoff relative to me instead of you then I should choose the maximin value and you should choose the minimax value.
Thus, this matrix is not equivalent to the turn-taking game, but rather represents a short-hand notation for what payoffs result when a sequence of actions are taken.
students.cs.byu.edu /~cs670ta/Lectures/Minimax.html   (1818 words)

 Replicator equation - Wikipedia, the free encyclopedia
Complete mixing: It is assumed that all individuals in the population have an equal chance to meet each other in a game to determine their interdependent payoff or fitness
Strategies breed true: Strategies are assumed to be inherited into the next population dependent on their expected payoff
payoff matrix defining the payoffs in the game.
www.wikipedia.org /wiki/Replicator_equation   (308 words)

 Payoff Matrix & Decision Rule   (Site not responding. Last check: 2007-10-06)
At the time the decision must be made the decision maker assumes that one of the columns represents the actual decision situation, but her or she does not know which column is the correct one.
The cells of the matrix represent payoffs that the decision maker would receive if he or she chose the action represented by a particular row and the actual state of nature were the one represented by a particular column.
If action A has a better payoff than action B under each individual state of nature, then we say that action B is stochastically dominated by action A.
www.whalens.org /Sofia/choice/matrix.htm   (396 words)

The payoff is the same as a hawk vs. a hawk, -25 each.
The payoff matrix containing the additional strategies will look as follows.
We see that none of these strategies are dominant over any of the others meaning that no strategy has payoffs greater than the corresponding payoffs in all other strategies.
www.stanford.edu /~alexch/hd_strategies.htm   (1030 words)

 STRATEGY 331 - Foundations of Strategy
Prepare a payoff matrix for the advertising game handed out in class (click here for a copy).
A payoff matrix is a simple representation of the payoffs to various combinations of strategies in two person games.
Recognize that the values within the payoff matrix change during the course of the game.
faculty.fuqua.duke.edu /~mlenox/fos/session7.htm   (340 words)

 Math Forum - Ask Dr. Math
Date: 06/04/2003 at 20:44:04 From: Elena Subject: Game theory problem: how to create a payoff matrix Shoe Town and Fancy Foot are both vying for more share of the market.
The payoffs in a game like this are usually measured in "utils" and not money or market share.
I was assuming that the payoff in market share was worth any cost to the players, but in reality losing 2% of market share might actually be worth less than $10,000.
mathforum.org /library/drmath/view/63044.html   (510 words)

 Payoff matrix Article, Payoffmatrix Information   (Site not responding. Last check: 2007-10-06)
Technically, matrix is used only in the case when there are two players and the payoff function for each can berepresented as a matrix.
In this game there are two players and the numerical payoff for each player is thesentence (time in jail) measured in years of confinement; in this case, lower is better.
In this matrix, player 1's strategies are designated along the left hand column and 2's are the designated along the toprow.
www.anoca.org /player/game/payoff_matrix.html   (728 words)

 ESS site Template
If the payoff to the strategy vs. itself is greater than any other in the column, it is likely that that particular strategy is a pure ESS.
Notice that these reference payoffs for a particular strategy are on a diagonal from upper left to lower right; they are shown as darkened cells in the matrix below:
Thus, for the first column of the matrix above, A is common and B, C, and D are all assumed to be rare.
www.holycross.edu /departments/biology/kprestwi/behavior/ESS/matrix_shortcut.html   (567 words)

 [No title]
In each cell of the table, the first number is the Firm 1’s payoff, and the second number is the Firm 2’s payoff.
For example, if both firm chose to enter, we are in the top left cell of the payoff matrix, and Firm 1’s payoff is 20, and Firm 2’s payoff is 40.
The reduced payoff matrix would be as follows: A very difficult problem - not!Firm 1EnterFirm 2Enter20, 40Not Enter100, 0 If Firm 2 enters, it gets a payoff of 40.
www.public.asu.edu /~lukashii/ecn112/gamehlp1.doc   (772 words)

 Payoff Matrix   (Site not responding. Last check: 2007-10-06)
Matrix which gives the possible outcome of a two-person
Matrix in order to determine optimal strategies is the aim of
The so-called ``augmented'' payoff matrix is defined as follows:
www.math.sdu.edu.cn /mathency/math/p/p145.htm   (29 words)

 Strategies of Play
Whereas the maximax principle is ultra-optimistic, expecting the best possible payoff, the maximin is ultra-pessimistic, expecting the worst possible payoff.
By this matrix, if the Japanese chose to take the southern route while the Allies decided to focus their recon planes in the north, the convoy would be bombed for 2 days.
The previous matrix had a saddle point, meaning that both the Japanese and the Allies settled on the (North, North) square as the best outcome for both of them.
cse.stanford.edu /classes/sophomore-college/projects-98/game-theory/Minimax.html   (1470 words)

 ARS | Publication request: The Impact Matrix Approach and Decision Rules to Enhance Index Dimensionality, Flexibility ...
We adapt the payoff matrix approach from financial risk to develop a comparable framework for ecosystem risk.
The payoff matrix utilizes the probability function for management alternatives to represent multiple pieces of information at one time (e.g.
Our proposal is to create an "impact matrix" that uses the same concepts as a payoff matrix.
www.ars.usda.gov /research/publications/publications.htm?SEQ_NO_115=124631   (623 words)

 Conservation Ecology: Incentive Systems that Support Sustainability: A First Nations Example: Appendix 2   (Site not responding. Last check: 2007-10-06)
But the payoff structure of the game is set up in such a way that neither player will cooperate unless some condition outside of the game provides inducement to do so.
In abstract form, the payoff matrix for the prisoner's dilemma can be written as follows in Table 1.
If we interpret each of the payoffs as the surplus received by each player from an economic activity, such as harvesting salmon, then imposition of a symmetric generosity rule changes the payoff matrix to the contents of Table 3:
sunsite.wits.ac.za /eco/vol2/iss2/art11/append2.html   (527 words)

The payoff form is a six by six matrix where each cell represents the number of clips to be gained (or lost).
The payoff matrix is a six by six matrix of numbers.
Consequently, residents must add or substract the new payoffs from their prior payoff matrix in order to have an accurate set of payoffs.
www.bsos.umd.edu /achen/ursp100/html/eol_rule.htm   (714 words)

 Introduction to Game Theory -- simple, two-strategy examples
Thus, E(C,C) is the payoff to a Call strategist (see note) when engaged in a contest with another Call strategist; E(S, C) is the payoff to a Satellite strategist when in a contest with a caller.
Thus, as long as the two strategies do not have identical payoffs (see this note to understand what would happen if both did have identical payoffs), then at some frequencies one will be more fit while at other frequencies the other strategy will be more fit.
Sometimes students who are familiar with basic probability and population biology assume that the frequency of a particular payoff equals a term in a binomial expansion of the strategy frequencies.
www.holycross.edu /departments/biology/kprestwi/behavior/ESS/games_intro.html   (7040 words)

 XS Mechanics   (Site not responding. Last check: 2007-10-06)
The rules for scoring the matrix and backtracking aren't too difficult to explain, but we needn't concern ourselves with them.
The entries in the score matrix are cells.
references are used to backtrack through the matrix after all the scores are filled in.
world.std.com /~swmcd/steven/perl/pm/xs/nw/NW.html   (1783 words)

 Thoughts about the Prisoner's Dilemma   (Site not responding. Last check: 2007-10-06)
Unfortunately for you, he is equally self interested, and will choose to always betray as well, so you're both stuck with a payoff of 3.
In that case the constant is 0, and the payoff matrix is uneffected.
It is important to note that the altered payoffs are not what the caring player actually receives.
www-users.cs.umn.edu /~damer/prisdilemma.html   (487 words)

 Two firms are exploring possibilities of entering a market for a new product
Both firms kno w the (new) payoff matrix, and each of the firms can figure out if the other has any dominant strategies.
After the transfer payment, the payoffs to this strategy in the collusive situation are: Firm 1 gets 40 and Firm 2 gets 60, which is be tter than 20 and 40 respectively they were making in the dominant strategy equilibrium.
Then it gets a payof f of 100, defrays the cost of 70 with it, and left with a residual payoff of 30, which is still larger than 20 it would have gotten had the Firm 2 been able to enter the market).
www.public.asu.edu /~lukashii/ecn112/gamehlp1.html   (1435 words)

 PayoffMatrix (PrisonerBot)
tells the payoff for each player for every permutation of player choices.
The "canonical" payoff matrix where: Mutual cooperation yields 3 points for each player.
an int array of length 2 where the first element is the payoff for the first player and the second element is the payoff for the second player
www.bolinfest.com /prisonerbot/api/com/bolinfest/prisonerbot/PayoffMatrix.html   (139 words)

 Math 130 - Finite Math   (Site not responding. Last check: 2007-10-06)
Of the land that was neither agricultural nor urban, 10% has been converted to agricultural land, 5% has been urbanized, and the remaining 85% remains unchanged.
Construct the transition matrix for the Markov chain that describes the shift in land use within the state.
Also compute the expected payoff f of each matrix game if the row and column players adopt their optimal strategies and determine which player it favors, if any.
math.boisestate.edu /~tconklin/TakehomeChap9.html   (665 words)

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