3444:
460:
91:, a normal-form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, whereas a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from 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
199:
189:
The matrix provided is a normal-form representation of a game in which players move simultaneously (or at least do not observe the other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For example, if player 1 plays top and player 2
423:, we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for a long time. However, if they both defect, they will both be locked up for a shorter time. One can determine that
99:—often cardinal in the normal-form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output.
190:
plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).
983:
221:(where the payoffs do not depend on which player chooses each action) are represented with only one payoff. This is the payoff for the row player. For example, the payoff matrices on the right and left below represent the same game.
549:). The above matrix does not represent the game in which player 1 moves first, observed by player 2, and then player 2 moves, because it does not specify each of player 2's strategies in this case. In order to represent this
1058:
988:
whose intended interpretation is the award given to a single player at the outcome of the game. Accordingly, to completely specify a game, the payoff function has to be specified for each player in the player set
886:
1276:
1193:
346:
The topological space of games with related payoff matrices can also be mapped, with adjacent games having the most similar matrices. This shows how incremental incentive changes can change the game.
782:
431:. One must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. This shows that no matter what the column player chooses, the row player does better by choosing
679:
1114:
435:. Similarly, one compares the second payoff in each row; again 0 > −1 and −2 > −5. This shows that no matter what row does, column does better by choosing
905:
553:
we must specify all of player 2's actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before,
76:, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable
463:
Both extensive and normal-form illustration of a sequential game with subgame imperfect and perfect Nash equilibria marked with red and blue respectively.
3102:
1013:
793:
3316:
2535:
3407:
1208:
1125:
1461:
1392:
1332:
707:
3326:
3092:
2360:
2177:
1712:
1510:
31:
3480:
1996:
1815:
1416:
1367:
1311:
1617:
3127:
2086:
2674:
1956:
1627:
1795:
624:
2137:
1555:
1530:
69:
17:
1436:, John Wiley Science Editions, 1964. Which was originally published in 1944 by Princeton University Press.
2891:
2528:
2487:
1913:
1667:
1657:
1592:
58:
1340:
2966:
1707:
1687:
1069:
545:
These matrices only represent games in which moves are simultaneous (or, more generally, information is
3122:
2644:
2421:
2172:
2142:
1800:
1642:
1637:
1384:
3226:
3097:
3011:
2457:
2380:
2116:
1672:
1597:
1454:
3331:
3221:
2929:
2609:
2472:
2205:
2091:
1888:
1682:
1500:
2275:
3366:
3295:
3177:
3037:
2634:
2521:
2477:
2076:
2046:
1702:
1490:
3236:
2819:
2624:
2502:
2482:
2462:
2411:
2081:
1986:
1845:
1790:
1722:
1692:
1612:
1540:
77:
561:. Unlike before he has four strategies, contingent on player 1's actions. The strategies are:
3182:
2919:
2769:
2764:
2599:
2574:
2569:
1961:
1946:
1520:
420:
203:
3376:
2734:
2564:
2544:
2295:
2280:
2167:
2162:
2066:
2051:
2016:
1981:
1580:
1525:
1447:
84:
65:
978:{\displaystyle u_{i}:S_{1}\times S_{2}\times \ldots \times S_{I}\rightarrow \mathbb {R} .}
8:
3397:
3371:
2949:
2754:
2744:
2452:
2071:
2021:
1858:
1785:
1765:
1622:
1505:
546:
88:
54:
2111:
3448:
3402:
3392:
3346:
3341:
3270:
3206:
3072:
2809:
2804:
2739:
2729:
2594:
2431:
2290:
2121:
2101:
1951:
1830:
1735:
1662:
1607:
416:
3459:
3443:
3246:
3241:
3231:
3211:
3172:
3167:
2996:
2991:
2976:
2971:
2962:
2957:
2904:
2799:
2749:
2694:
2664:
2659:
2639:
2629:
2589:
2416:
2385:
2340:
2235:
2106:
2061:
2036:
1966:
1840:
1770:
1760:
1652:
1602:
1550:
1429:
1412:
1388:
1363:
1328:
1307:
459:
3454:
3422:
3351:
3290:
3285:
3265:
3201:
3107:
3077:
3062:
3042:
2981:
2934:
2909:
2899:
2870:
2789:
2784:
2759:
2689:
2669:
2579:
2559:
2497:
2492:
2426:
2390:
2370:
2330:
2300:
2255:
2210:
2195:
2152:
2006:
1647:
1584:
1570:
1425:
1356:
594:
In order for a game to be in normal form, we are provided with the following data:
440:
92:
73:
3047:
3152:
3087:
3067:
3052:
3032:
3016:
2914:
2845:
2835:
2794:
2679:
2649:
2395:
2355:
2310:
2225:
2220:
1941:
1893:
1780:
1545:
1515:
1485:
550:
96:
2260:
3412:
3356:
3336:
3321:
3280:
3157:
3117:
3082:
3006:
2945:
2924:
2865:
2855:
2840:
2774:
2719:
2709:
2704:
2614:
2335:
2325:
2315:
2250:
2240:
2230:
2215:
2011:
1991:
1976:
1971:
1931:
1898:
1883:
1878:
1868:
1677:
1347:
1295:
218:
211:
1400:
202:
A partial topology of two-player, two-strategy games, including such games as
3474:
3417:
3275:
3216:
3147:
3137:
3132:
3057:
2986:
2860:
2850:
2779:
2699:
2684:
2619:
2375:
2365:
2320:
2305:
2285:
2056:
2031:
1903:
1873:
1863:
1850:
1755:
1697:
1632:
1565:
1399:. A comprehensive reference from a computational perspective; see Chapter 3.
1351:
614:
3300:
3257:
3162:
2875:
2814:
2724:
2604:
2350:
2345:
2200:
1775:
35:
3142:
3112:
2880:
2714:
2584:
2467:
2270:
2265:
2245:
2041:
2026:
1835:
1805:
1740:
1730:
1560:
1495:
1471:
1299:
419:, and it is usually used to illustrate this concept. For example, in the
42:
1439:
1380:
Multiagent
Systems: Algorithmic, Game-Theoretic, and Logical Foundations
1053:{\displaystyle \mathrm {T} =\langle I,\mathbf {S} ,\mathbf {u} \rangle }
3193:
2654:
2096:
1750:
3427:
3001:
2001:
1921:
1745:
207:
1324:
Essentials of Game Theory: A Concise, Multidisciplinary
Introduction
881:{\displaystyle s_{1}\in S_{1},s_{2}\in S_{2},\ldots ,s_{I}\in S_{I}}
3361:
2436:
1936:
2513:
2157:
2147:
1825:
198:
1378:
1322:
1926:
699:
586:
On the right is the normal-form representation of this game.
68:. While this approach can be of greater use in identifying
1271:{\displaystyle \mathbf {u} =\{u_{1},u_{2},\ldots ,u_{I}\}}
1188:{\displaystyle \mathbf {S} =\{S_{1},S_{2},\ldots ,S_{I}\}}
32:
Storytelling game § Alternate form role-playing games
30:"Matrix game" redirects here. For Chris Engle's game, see
1202:-tuple of pure strategy sets, one for each player, and
694:
is an association of strategies to players, that is an
777:{\displaystyle {\vec {s}}=(s_{1},s_{2},\ldots ,s_{I})}
1211:
1128:
1072:
1016:
908:
796:
710:
627:
1327:. San Rafael, CA: Morgan & Claypool Publishers.
80:, and their corresponding payoffs, for each player.
1376:
1355:
1320:
1270:
1187:
1108:
1052:
977:
880:
776:
673:
454:
3472:
1294:
581:Right if player 1 plays Top and Right otherwise
576:Right if player 1 plays Top and Left otherwise
571:Left if player 1 plays Top and Right otherwise
2529:
1455:
566:Left if player 1 plays Top and Left otherwise
415:The payoff matrix facilitates elimination of
1265:
1220:
1182:
1137:
1103:
1079:
1047:
1025:
665:
641:
64:, but rather represent the game by way of a
3103:Fundamental (linear differential equation)
2536:
2522:
1462:
1448:
1377:Shoham, Yoav; Leyton-Brown, Kevin (2009).
1346:
1321:Leyton-Brown, Kevin; Shoham, Yoav (2008).
1469:
968:
1339:. An 88-page mathematical introduction;
674:{\displaystyle S_{i}=\{1,2,\ldots ,k\}.}
458:
197:
193:
3408:Matrix representation of conic sections
1406:
354:
27:Representation of a game in game theory
14:
3473:
601:of players, each player is denoted by
589:
349:
57:, normal-form representations are not
2517:
1443:
1434:Theory of games and Economic Behavior
1109:{\displaystyle I=\{1,2,\ldots ,I\}}
24:
2543:
1511:First-player and second-player win
1018:
25:
3492:
467:
359:
286:
228:
3442:
1618:Coalition-proof Nash equilibrium
1213:
1130:
1043:
1035:
3310:Used in science and engineering
455:Sequential games in normal form
439:. This demonstrates the unique
2553:Explicitly constrained entries
1628:Evolutionarily stable strategy
964:
771:
726:
717:
13:
1:
3327:Fundamental (computer vision)
1556:Simultaneous action selection
1288:
102:
70:strictly dominated strategies
2488:List of games in game theory
1668:Quantal response equilibrium
1658:Perfect Bayesian equilibrium
1593:Bayes correlated equilibrium
1285:-tuple of payoff functions.
538:
535:
532:
529:
519:
516:
513:
510:
7:
3093:Duplication and elimination
2892:eigenvalues or eigenvectors
1957:Optional prisoner's dilemma
1688:Self-confirming equilibrium
10:
3497:
3026:With specific applications
2655:Discrete Fourier Transform
2422:Principal variation search
2138:Aumann's agreement theorem
1801:Strategy-stealing argument
1713:Trembling hand equilibrium
1643:Markov perfect equilibrium
1638:Mertens-stable equilibrium
1385:Cambridge University Press
29:
3436:
3385:
3317:Cabibbo–Kobayashi–Maskawa
3309:
3255:
3191:
3025:
2944:Satisfying conditions on
2943:
2889:
2828:
2552:
2458:Combinatorial game theory
2445:
2404:
2186:
2130:
2117:Princess and monster game
1912:
1814:
1721:
1673:Quasi-perfect equilibrium
1598:Bayesian Nash equilibrium
1579:
1478:
427:is strictly dominated by
162:
137:
130:
125:
34:. For the publisher, see
3481:Game theory game classes
2473:Evolutionary game theory
2206:Antoine Augustin Cournot
2092:Guess 2/3 of the average
1889:Strictly determined game
1683:Satisfaction equilibrium
1501:Escalation of commitment
1409:Evolutionary Game Theory
1401:Downloadable free online
2675:Generalized permutation
2478:Glossary of game theory
2077:Stackelberg competition
1703:Strong Nash equilibrium
3449:Mathematics portal
2503:Tragedy of the commons
2483:List of game theorists
2463:Confrontation analysis
2173:Sprague–Grundy theorem
1693:Sequential equilibrium
1613:Correlated equilibrium
1362:. Dover Publications.
1272:
1189:
1110:
1054:
979:
882:
778:
675:
597:There is a finite set
464:
361:The Prisoner's Dilemma
214:
49:is a description of a
2276:Jean-François Mertens
1343:at many universities.
1273:
1190:
1119:is a set of players,
1111:
1055:
980:
883:
779:
690:pure strategy profile
676:
462:
201:
194:Other representations
2405:Search optimizations
2281:Jennifer Tour Chayes
2168:Revelation principle
2163:Purification theorem
2102:Nash bargaining game
2067:Bertrand competition
2052:El Farol Bar problem
2017:Electronic mail game
1982:Lewis signaling game
1526:Hierarchy of beliefs
1407:Weibull, J. (1996).
1209:
1126:
1070:
1014:
906:
794:
708:
625:
417:dominated strategies
355:Dominated strategies
3398:Linear independence
2645:Diagonally dominant
2453:Bounded rationality
2072:Cournot competition
2022:Rock paper scissors
1997:Battle of the sexes
1987:Volunteer's dilemma
1859:Perfect information
1786:Dominant strategies
1623:Epsilon-equilibrium
1506:Extensive-form game
1358:Games and Decisions
1005:game in normal form
590:General formulation
471:
363:
350:Uses of normal form
290:
232:
109:
108:A normal-form game
89:perfect information
83:In static games of
3403:Matrix exponential
3393:Jordan normal form
3227:Fisher information
3098:Euclidean distance
3012:Totally unimodular
2432:Paranoid algorithm
2412:Alpha–beta pruning
2291:John Maynard Smith
2122:Rendezvous problem
1962:Traveler's dilemma
1952:Gift-exchange game
1947:Prisoner's dilemma
1864:Large Poisson game
1831:Bargaining problem
1736:Backward induction
1708:Subgame perfection
1663:Proper equilibrium
1268:
1185:
1106:
1050:
975:
878:
774:
671:
465:
421:prisoner's dilemma
215:
204:Prisoner's dilemma
107:
3468:
3467:
3460:Category:Matrices
3332:Fuzzy associative
3222:Doubly stochastic
2930:Positive-definite
2610:Block tridiagonal
2511:
2510:
2417:Aspiration window
2386:Suzanne Scotchmer
2341:Oskar Morgenstern
2236:Donald B. Gillies
2178:Zermelo's theorem
2107:Induction puzzles
2062:Fair cake-cutting
2037:Public goods game
1967:Coordination game
1841:Intransitive game
1771:Forward induction
1653:Pareto efficiency
1633:Gibbs equilibrium
1603:Berge equilibrium
1551:Simultaneous game
1394:978-0-521-89943-7
1334:978-1-59829-593-1
720:
543:
542:
469:A sequential game
443:of this game is (
413:
412:
344:
343:
340:
339:
282:
281:
187:
186:
16:(Redirected from
3488:
3455:List of matrices
3447:
3446:
3423:Row echelon form
3367:State transition
3296:Seidel adjacency
3178:Totally positive
3038:Alternating sign
2635:Complex Hadamard
2538:
2531:
2524:
2515:
2514:
2498:Topological game
2493:No-win situation
2391:Thomas Schelling
2371:Robert B. Wilson
2331:Merrill M. Flood
2301:John von Neumann
2211:Ariel Rubinstein
2196:Albert W. Tucker
2047:War of attrition
2007:Matching pennies
1648:Nash equilibrium
1571:Mechanism design
1536:Normal-form game
1491:Cooperative game
1464:
1457:
1450:
1441:
1440:
1422:
1398:
1373:
1361:
1338:
1317:
1277:
1275:
1274:
1269:
1264:
1263:
1245:
1244:
1232:
1231:
1216:
1194:
1192:
1191:
1186:
1181:
1180:
1162:
1161:
1149:
1148:
1133:
1115:
1113:
1112:
1107:
1059:
1057:
1056:
1051:
1046:
1038:
1021:
984:
982:
981:
976:
971:
963:
962:
944:
943:
931:
930:
918:
917:
897:
896:
887:
885:
884:
879:
877:
876:
864:
863:
845:
844:
832:
831:
819:
818:
806:
805:
783:
781:
780:
775:
770:
769:
751:
750:
738:
737:
722:
721:
713:
692:
691:
680:
678:
677:
672:
637:
636:
472:
466:
441:Nash equilibrium
364:
358:
291:
285:
233:
227:
224:
223:
183:
179:
174:
170:
165:
158:
154:
149:
145:
140:
133:
128:
122:
116:
110:
106:
21:
3496:
3495:
3491:
3490:
3489:
3487:
3486:
3485:
3471:
3470:
3469:
3464:
3441:
3432:
3381:
3305:
3251:
3187:
3021:
2939:
2885:
2824:
2625:Centrosymmetric
2548:
2542:
2512:
2507:
2441:
2427:max^n algorithm
2400:
2396:William Vickrey
2356:Reinhard Selten
2311:Kenneth Binmore
2226:David K. Levine
2221:Daniel Kahneman
2188:
2182:
2158:Negamax theorem
2148:Minimax theorem
2126:
2087:Diner's dilemma
1942:All-pay auction
1908:
1894:Stochastic game
1846:Mean-field game
1817:
1810:
1781:Markov strategy
1717:
1583:
1575:
1546:Sequential game
1531:Information set
1516:Game complexity
1486:Congestion game
1474:
1468:
1419:
1395:
1370:
1335:
1314:
1291:
1259:
1255:
1240:
1236:
1227:
1223:
1212:
1210:
1207:
1206:
1176:
1172:
1157:
1153:
1144:
1140:
1129:
1127:
1124:
1123:
1071:
1068:
1067:
1042:
1034:
1017:
1015:
1012:
1011:
1007:is a structure
967:
958:
954:
939:
935:
926:
922:
913:
909:
907:
904:
903:
895:payoff function
894:
893:
872:
868:
859:
855:
840:
836:
827:
823:
814:
810:
801:
797:
795:
792:
791:
765:
761:
746:
742:
733:
729:
712:
711:
709:
706:
705:
689:
688:
632:
628:
626:
623:
622:
615:pure strategies
592:
551:sequential game
481:
479:
477:
457:
373:
371:
369:
357:
352:
300:
298:
296:
242:
240:
238:
219:symmetric games
196:
181:
177:
172:
168:
163:
156:
152:
147:
143:
138:
131:
126:
123:
120:
119:
117:
114:
105:
97:ordinal utility
74:Nash equilibria
39:
28:
23:
22:
15:
12:
11:
5:
3494:
3484:
3483:
3466:
3465:
3463:
3462:
3457:
3452:
3437:
3434:
3433:
3431:
3430:
3425:
3420:
3415:
3413:Perfect matrix
3410:
3405:
3400:
3395:
3389:
3387:
3383:
3382:
3380:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3319:
3313:
3311:
3307:
3306:
3304:
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3268:
3262:
3260:
3253:
3252:
3250:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3198:
3196:
3189:
3188:
3186:
3185:
3183:Transformation
3180:
3175:
3170:
3165:
3160:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3090:
3085:
3080:
3075:
3070:
3065:
3060:
3055:
3050:
3045:
3040:
3035:
3029:
3027:
3023:
3022:
3020:
3019:
3014:
3009:
3004:
2999:
2994:
2989:
2984:
2979:
2974:
2969:
2960:
2954:
2952:
2941:
2940:
2938:
2937:
2932:
2927:
2922:
2920:Diagonalizable
2917:
2912:
2907:
2902:
2896:
2894:
2890:Conditions on
2887:
2886:
2884:
2883:
2878:
2873:
2868:
2863:
2858:
2853:
2848:
2843:
2838:
2832:
2830:
2826:
2825:
2823:
2822:
2817:
2812:
2807:
2802:
2797:
2792:
2787:
2782:
2777:
2772:
2770:Skew-symmetric
2767:
2765:Skew-Hermitian
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2712:
2707:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2600:Block-diagonal
2597:
2592:
2587:
2582:
2577:
2575:Anti-symmetric
2572:
2570:Anti-Hermitian
2567:
2562:
2556:
2554:
2550:
2549:
2541:
2540:
2533:
2526:
2518:
2509:
2508:
2506:
2505:
2500:
2495:
2490:
2485:
2480:
2475:
2470:
2465:
2460:
2455:
2449:
2447:
2443:
2442:
2440:
2439:
2434:
2429:
2424:
2419:
2414:
2408:
2406:
2402:
2401:
2399:
2398:
2393:
2388:
2383:
2378:
2373:
2368:
2363:
2361:Robert Axelrod
2358:
2353:
2348:
2343:
2338:
2336:Olga Bondareva
2333:
2328:
2326:Melvin Dresher
2323:
2318:
2316:Leonid Hurwicz
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2251:Harold W. Kuhn
2248:
2243:
2241:Drew Fudenberg
2238:
2233:
2231:David M. Kreps
2228:
2223:
2218:
2216:Claude Shannon
2213:
2208:
2203:
2198:
2192:
2190:
2184:
2183:
2181:
2180:
2175:
2170:
2165:
2160:
2155:
2153:Nash's theorem
2150:
2145:
2140:
2134:
2132:
2128:
2127:
2125:
2124:
2119:
2114:
2109:
2104:
2099:
2094:
2089:
2084:
2079:
2074:
2069:
2064:
2059:
2054:
2049:
2044:
2039:
2034:
2029:
2024:
2019:
2014:
2012:Ultimatum game
2009:
2004:
1999:
1994:
1992:Dollar auction
1989:
1984:
1979:
1977:Centipede game
1974:
1969:
1964:
1959:
1954:
1949:
1944:
1939:
1934:
1932:Infinite chess
1929:
1924:
1918:
1916:
1910:
1909:
1907:
1906:
1901:
1899:Symmetric game
1896:
1891:
1886:
1884:Signaling game
1881:
1879:Screening game
1876:
1871:
1869:Potential game
1866:
1861:
1856:
1848:
1843:
1838:
1833:
1828:
1822:
1820:
1812:
1811:
1809:
1808:
1803:
1798:
1796:Mixed strategy
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1743:
1738:
1733:
1727:
1725:
1719:
1718:
1716:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1680:
1678:Risk dominance
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1640:
1635:
1630:
1625:
1620:
1615:
1610:
1605:
1600:
1595:
1589:
1587:
1577:
1576:
1574:
1573:
1568:
1563:
1558:
1553:
1548:
1543:
1538:
1533:
1528:
1523:
1521:Graphical game
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1482:
1480:
1476:
1475:
1467:
1466:
1459:
1452:
1444:
1438:
1437:
1430:O. Morgenstern
1426:J. von Neumann
1423:
1417:
1404:
1393:
1374:
1368:
1344:
1333:
1318:
1312:
1290:
1287:
1279:
1278:
1267:
1262:
1258:
1254:
1251:
1248:
1243:
1239:
1235:
1230:
1226:
1222:
1219:
1215:
1196:
1195:
1184:
1179:
1175:
1171:
1168:
1165:
1160:
1156:
1152:
1147:
1143:
1139:
1136:
1132:
1117:
1116:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1061:
1060:
1049:
1045:
1041:
1037:
1033:
1030:
1027:
1024:
1020:
993:= {1, 2, ...,
986:
985:
974:
970:
966:
961:
957:
953:
950:
947:
942:
938:
934:
929:
925:
921:
916:
912:
899:is a function
889:
888:
875:
871:
867:
862:
858:
854:
851:
848:
843:
839:
835:
830:
826:
822:
817:
813:
809:
804:
800:
785:
784:
773:
768:
764:
760:
757:
754:
749:
745:
741:
736:
732:
728:
725:
719:
716:
684:
683:
682:
681:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
635:
631:
605:. Each player
591:
588:
584:
583:
578:
573:
568:
541:
540:
537:
534:
531:
528:
522:
521:
518:
515:
512:
509:
503:
502:
497:
492:
487:
482:
478:
475:
456:
453:
411:
410:
407:
404:
398:
397:
394:
391:
385:
384:
379:
374:
370:
367:
356:
353:
351:
348:
342:
341:
338:
337:
334:
331:
325:
324:
321:
318:
312:
311:
306:
301:
297:
294:
283:
280:
279:
276:
273:
267:
266:
263:
260:
254:
253:
248:
243:
239:
236:
195:
192:
185:
184:
175:
166:
160:
159:
150:
141:
135:
134:
129:
124:
118:
113:
104:
101:
55:extensive form
26:
9:
6:
4:
3:
2:
3493:
3482:
3479:
3478:
3476:
3461:
3458:
3456:
3453:
3451:
3450:
3445:
3439:
3438:
3435:
3429:
3426:
3424:
3421:
3419:
3418:Pseudoinverse
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3390:
3388:
3386:Related terms
3384:
3378:
3377:Z (chemistry)
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3314:
3312:
3308:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3263:
3261:
3259:
3254:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3199:
3197:
3195:
3190:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3030:
3028:
3024:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2964:
2961:
2959:
2956:
2955:
2953:
2951:
2947:
2942:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2901:
2898:
2897:
2895:
2893:
2888:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2833:
2831:
2827:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2735:Pentadiagonal
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2565:Anti-diagonal
2563:
2561:
2558:
2557:
2555:
2551:
2546:
2539:
2534:
2532:
2527:
2525:
2520:
2519:
2516:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2484:
2481:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2450:
2448:
2446:Miscellaneous
2444:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2409:
2407:
2403:
2397:
2394:
2392:
2389:
2387:
2384:
2382:
2381:Samuel Bowles
2379:
2377:
2376:Roger Myerson
2374:
2372:
2369:
2367:
2366:Robert Aumann
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2321:Lloyd Shapley
2319:
2317:
2314:
2312:
2309:
2307:
2306:Kenneth Arrow
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2286:John Harsanyi
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2256:Herbert Simon
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2193:
2191:
2185:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2135:
2133:
2129:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2057:Fair division
2055:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2032:Dictator game
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1919:
1917:
1915:
1911:
1905:
1904:Zero-sum game
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1874:Repeated game
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1853:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1823:
1821:
1819:
1813:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1791:Pure strategy
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1757:
1756:De-escalation
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1728:
1726:
1724:
1720:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1698:Shapley value
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1604:
1601:
1599:
1596:
1594:
1591:
1590:
1588:
1586:
1582:
1578:
1572:
1569:
1567:
1566:Succinct game
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1483:
1481:
1477:
1473:
1465:
1460:
1458:
1453:
1451:
1446:
1445:
1442:
1435:
1431:
1427:
1424:
1420:
1418:0-262-23181-6
1414:
1411:. MIT Press.
1410:
1405:
1402:
1396:
1390:
1386:
1382:
1381:
1375:
1371:
1369:0-486-65943-7
1365:
1360:
1359:
1353:
1349:
1345:
1342:
1336:
1330:
1326:
1325:
1319:
1315:
1313:0-262-06141-4
1309:
1306:. MIT Press.
1305:
1301:
1297:
1296:Fudenberg, D.
1293:
1292:
1286:
1284:
1260:
1256:
1252:
1249:
1246:
1241:
1237:
1233:
1228:
1224:
1217:
1205:
1204:
1203:
1201:
1177:
1173:
1169:
1166:
1163:
1158:
1154:
1150:
1145:
1141:
1134:
1122:
1121:
1120:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1076:
1073:
1066:
1065:
1064:
1039:
1031:
1028:
1022:
1010:
1009:
1008:
1006:
1002:
998:
996:
992:
972:
959:
955:
951:
948:
945:
940:
936:
932:
927:
923:
919:
914:
910:
902:
901:
900:
898:
873:
869:
865:
860:
856:
852:
849:
846:
841:
837:
833:
828:
824:
820:
815:
811:
807:
802:
798:
790:
789:
788:
766:
762:
758:
755:
752:
747:
743:
739:
734:
730:
723:
714:
704:
703:
702:
701:
697:
693:
668:
662:
659:
656:
653:
650:
647:
644:
638:
633:
629:
621:
620:
619:
618:
617:
616:
612:
609:has a finite
608:
604:
600:
595:
587:
582:
579:
577:
574:
572:
569:
567:
564:
563:
562:
560:
556:
552:
548:
527:
524:
523:
508:
505:
504:
501:
498:
496:
493:
491:
488:
486:
483:
474:
473:
470:
461:
452:
450:
446:
442:
438:
434:
430:
426:
422:
418:
408:
405:
403:
400:
399:
395:
392:
390:
387:
386:
383:
380:
378:
375:
366:
365:
362:
347:
335:
332:
330:
327:
326:
322:
319:
317:
314:
313:
310:
307:
305:
302:
293:
292:
289:
284:
277:
274:
272:
269:
268:
264:
261:
259:
256:
255:
252:
249:
247:
244:
235:
234:
231:
226:
225:
222:
220:
213:
209:
205:
200:
191:
176:
167:
161:
151:
142:
136:
112:
111:
100:
98:
94:
90:
86:
81:
79:
75:
71:
67:
63:
60:
56:
52:
48:
44:
37:
33:
19:
18:Payoff matrix
3440:
3372:Substitution
3258:graph theory
2755:Quaternionic
2745:Persymmetric
2351:Peyton Young
2346:Paul Milgrom
2261:Hervé Moulin
2201:Amos Tversky
2143:Folk theorem
1854:-player game
1851:
1776:Grim trigger
1535:
1433:
1408:
1383:. New York:
1379:
1357:
1323:
1303:
1282:
1280:
1199:
1197:
1118:
1062:
1004:
1000:
999:
994:
990:
987:
892:
890:
786:
695:
687:
685:
610:
606:
602:
598:
596:
593:
585:
580:
575:
570:
565:
558:
554:
544:
525:
506:
500:Right, Right
499:
494:
489:
484:
468:
448:
444:
436:
432:
428:
424:
414:
401:
388:
381:
376:
360:
345:
328:
315:
308:
303:
287:
270:
257:
250:
245:
230:Both players
229:
216:
188:
82:
61:
50:
46:
40:
36:Matrix Games
3347:Hamiltonian
3271:Biadjacency
3207:Correlation
3123:Householder
3073:Commutation
2810:Vandermonde
2805:Tridiagonal
2740:Permutation
2730:Nonnegative
2715:Matrix unit
2595:Bisymmetric
2468:Coopetition
2271:Jean Tirole
2266:John Conway
2246:Eric Maskin
2042:Blotto game
2027:Pirate game
1836:Global game
1806:Tit for tat
1741:Bid shading
1731:Appeasement
1581:Equilibrium
1561:Solved game
1496:Determinacy
1479:Definitions
1472:game theory
1348:Luce, R. D.
1341:free online
1304:Game Theory
495:Right, Left
490:Left, Right
47:normal form
43:game theory
3247:Transition
3242:Stochastic
3212:Covariance
3194:statistics
3173:Symplectic
3168:Similarity
2997:Unimodular
2992:Orthogonal
2977:Involutory
2972:Invertible
2967:Projection
2963:Idempotent
2905:Convergent
2800:Triangular
2750:Polynomial
2695:Hessenberg
2665:Equivalent
2660:Elementary
2640:Copositive
2630:Conference
2590:Bidiagonal
2112:Trust game
2097:Kuhn poker
1766:Escalation
1761:Deterrence
1751:Cheap talk
1723:Strategies
1541:Preference
1470:Topics of
1352:Raiffa, H.
1300:Tirole, J.
1289:References
1001:Definition
787:such that
613:number of
485:Left, Left
103:An example
78:strategies
3428:Wronskian
3352:Irregular
3342:Gell-Mann
3291:Laplacian
3286:Incidence
3266:Adjacency
3237:Precision
3202:Centering
3108:Generator
3078:Confusion
3063:Circulant
3043:Augmented
3002:Unipotent
2982:Nilpotent
2958:Congruent
2935:Stieltjes
2910:Defective
2900:Companion
2871:Redheffer
2790:Symmetric
2785:Sylvester
2760:Signature
2690:Hermitian
2670:Frobenius
2580:Arrowhead
2560:Alternant
2296:John Nash
2002:Stag hunt
1746:Collusion
1250:…
1167:…
1095:…
1048:⟩
1026:⟨
965:→
952:×
949:…
946:×
933:×
866:∈
850:…
834:∈
808:∈
756:…
718:→
657:…
547:imperfect
425:Cooperate
389:Cooperate
377:Cooperate
208:Stag hunt
59:graphical
53:. Unlike
3475:Category
3256:Used in
3192:Used in
3153:Rotation
3128:Jacobian
3088:Distance
3068:Cofactor
3053:Carleman
3033:Adjugate
3017:Weighing
2950:inverses
2946:products
2915:Definite
2846:Identity
2836:Exchange
2829:Constant
2795:Toeplitz
2680:Hadamard
2650:Diagonal
2437:Lazy SMP
2131:Theorems
2082:Deadlock
1937:Checkers
1818:of games
1585:concepts
1354:(1989).
1302:(1991).
480:Player 1
476:Player 2
372:Player 1
368:Player 2
299:Player 1
295:Player 2
288:Just row
241:Player 1
237:Player 2
121:Player 1
115:Player 2
93:cardinal
85:complete
3357:Overlap
3322:Density
3281:Edmonds
3158:Seifert
3118:Hessian
3083:Coxeter
3007:Unitary
2925:Hurwitz
2856:Of ones
2841:Hilbert
2775:Skyline
2720:Metzler
2710:Logical
2705:Integer
2615:Boolean
2547:classes
2189:figures
1972:Chicken
1826:Auction
1816:Classes
1063:where:
520:−1, −1
517:−1, −1
409:−2, −2
393:−1, −1
217:Often,
212:Chicken
3276:Degree
3217:Design
3148:Random
3138:Payoff
3133:Moment
3058:Cartan
3048:Bézout
2987:Normal
2861:Pascal
2851:Lehmer
2780:Sparse
2700:Hollow
2685:Hankel
2620:Cauchy
2545:Matrix
1415:
1391:
1366:
1331:
1310:
1281:is an
1198:is an
526:Bottom
449:Defect
445:Defect
437:Defect
433:Defect
429:Defect
406:0, −5
402:Defect
396:−5, 0
382:Defect
210:, and
164:Bottom
66:matrix
62:per se
3337:Gamma
3301:Tutte
3163:Shear
2876:Shift
2866:Pauli
2815:Walsh
2725:Moore
2605:Block
1927:Chess
1914:Games
700:tuple
559:Right
539:3, 4
536:0, 0
533:3, 4
530:0, 0
514:4, 3
511:4, 3
278:2, 2
275:2, 0
265:0, 2
262:3, 3
132:Right
3143:Pick
3113:Gram
2881:Zero
2585:Band
1608:Core
1428:and
1413:ISBN
1389:ISBN
1364:ISBN
1329:ISBN
1308:ISBN
1003:: A
557:and
555:Left
329:Hare
316:Stag
309:Hare
304:Stag
271:Hare
258:Stag
251:Hare
246:Stag
127:Left
72:and
51:game
3232:Hat
2965:or
2948:or
2187:Key
997:}.
507:Top
451:).
139:Top
95:or
41:In
3477::
1922:Go
1432:,
1387:.
1350:;
1298:;
891:A
686:A
447:,
336:2
333:2
323:0
320:3
206:,
180:,
171:,
157:−1
155:,
153:−1
146:,
87:,
45:,
3362:S
2820:Z
2537:e
2530:t
2523:v
1852:n
1463:e
1456:t
1449:v
1421:.
1403:.
1397:.
1372:.
1337:.
1316:.
1283:I
1266:}
1261:I
1257:u
1253:,
1247:,
1242:2
1238:u
1234:,
1229:1
1225:u
1221:{
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1214:u
1200:I
1183:}
1178:I
1174:S
1170:,
1164:,
1159:2
1155:S
1151:,
1146:1
1142:S
1138:{
1135:=
1131:S
1104:}
1101:I
1098:,
1092:,
1089:2
1086:,
1083:1
1080:{
1077:=
1074:I
1044:u
1040:,
1036:S
1032:,
1029:I
1023:=
1019:T
995:I
991:I
973:.
969:R
960:I
956:S
941:2
937:S
928:1
924:S
920::
915:i
911:u
874:I
870:S
861:I
857:s
853:,
847:,
842:2
838:S
829:2
825:s
821:,
816:1
812:S
803:1
799:s
772:)
767:I
763:s
759:,
753:,
748:2
744:s
740:,
735:1
731:s
727:(
724:=
715:s
698:-
696:I
669:.
666:}
663:k
660:,
654:,
651:2
648:,
645:1
642:{
639:=
634:i
630:S
611:k
607:i
603:i
599:I
182:4
178:3
173:0
169:0
148:3
144:4
38:.
20:)
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