2 X 2 Games1. Visualizing the Adjacent Possible in the Topology of Two-player, Two-strategy Games Families Harmonious

Adjacent games are neighbors by payoff swaps 1.Win-win 4,4 Stag Hunt Game Column payoffs Graph 1!2 swaps form tiles of 4 games Tile Layers 2.Biased 4,3 Battle

Pd 1 4

3 3 Row payoffs 2!3 swaps join tiles into 4 layers 4 #

4#

4 1 Self-serving Altruistic

2 2

4 1 Nash equilibrium 3!4 swaps link layers 4

#4

# 3.Second Best 3,3 CC9966

111Prisoner's (Maximin for cyclic) Layers differ by alignment of 4s 4 4

#4

4# 4.Unfair 4,2 Winner Loser

Dilemma Pareto-deficient Inducements Each layer is a torus, table is a torus 3 2 5.PD Family Tragic 3,2Pareto-optimal outcomes in bold font Layers scrolled to center Prisoner's Dilemma Pd 2,2 Alibi 3,2

4 Chart 6.Cyclic 1Hg 2

33

4 Sd 2 2

3 4 QQtt 2

13

4 Bc 2 1

3 4 Ic 2

23

4 En 2 3

3 4 Cb 2

43

3 Bu 2 4

3 2 UtUt 2

43

1 Sb 2 4

3 1 Ab 2

43

2 Ch 2 4

3 3 2 3

1 1

4 2

1 1

4 3

1 2

4 3

1 3

4 2

1 3

4 1

1 2

4 1

1 2

4 1

1 3

4 1

1 3

4 2

1 2

4 3

1 1

4 3

1 1

4 2 1 4

421Hegemony 426SamaritanD 425QuasiAltruistType424BiasedCycle 423InferiorCycle 422Endless 121Called Bluff 126Bully 125Unfair Type 124Skewed Ba. 123Asym Battle 122Chicken 2Sn 3

32

4 Ge 3 2

2 4 Ge 3

12

4 Qc 3 1

2 4 Tc 3

22

4 In 3 3

2 4 Pn 3

42

3 Pr 3 4

2 2 Pr 3

42

1 Qb 3 4

2 1 Ba 3

42

2 Ab 3 4

2 3 3 2

1 1

4 2

1 1

4 3

1 2

4 3

1 3

4 2

1 3

4 1

1 2

4 1

1 2

4 1

1 3

4 1

1 3

4 2

1 2

4 3

1 1

4 3

1 1

4 2 1 4

431Samson 436Generous 435Generous 434Quasi Cycle 433TragicCycle 432Inspector 131Patron 136Protector 135Protector 134QuasiBattle 133Battle 132Asym Battle 3Dh 3

31

4 Ge 3 2

1 4 Ge 3

11

4 Pc 3 1

1 4 Fs 3

21

4Mc 3 3

1 4 Pn 3

41

3 Pr 3 4

1 2 Pr 3

41

1 Hr 3 4

1 1 Qb 3

41

2 Sb 3 4

1 3 3 1

2 1

4 2

2 1

4 3

2 2

4 3

2 3

4 2

2 3

4 1

2 2

4 1

2 2

4 1

2 3

4 1

2 3

4 2

2 2

4 3

2 1

4 3

2 1

4 2 2 4

441Delilah 446Generous 445Generous 444PursuitCycle 443Fixed Sum 442MissileCrisis 141Patron 146Protector 145Protector 144Hero 143QuasiBattle 142Skewed Ba. 4Hs 2

31

4 Be 2 2

1 4 Su 2

11

4 Se 2 1

1 4 Se 2

21

4 Bb 2 3

1 4 Tr 2

41

3 Ad 2 4

1 2 De 2

41

1 Pr 2 4

1 1 Pr 2

41

2 UtUt 2 4

1 3 2 1

3 1

4 2

3 1

4 3

3 2

4 3

3 3

4 2

3 3

4 1

3 2

4 1

3 2

4 1

3 3

4 1

3 3

4 2

3 2

4 3

3 1

4 3

3 1

4 2 3 4

451Hostage 456Benevolent 455Subsym Be 454SecondBest 453SecondBest 452Big Bully 151Tragedy 156Asym Del. 155P. Delight 154Protector 153Protector 152Unfair Type 5MtMt 1

32

4 BtBt 1 2

2 4 Be 1

12

4 Se 1 1

2 4 Se 1

22

4 Ha 1 3

2 4 Tc 1

42

3 Dl 1 4

2 2 Ad 1

42

1 Pr 1 4

2 1 Pr 1

42

2 Bu 1 4

2 3 1 2

3 1

4 2

3 1

4 3

3 2

4 3

3 3

4 2

3 3

4 1

3 2

4 1

3 2

4 1 3

34

1 3 3

4 2

3 2

4 3

3 1

4 3

3 1

4 2 3 4

461BlackmailTy. 466Benev.Type 465Benevolent 464SecondBest 463SecondBest 462Hamlet 161TotalConflict 166Deadlock 165Asym Del. 164Protector 163Protector 162Bully 6HtHt 1

33

4 AtAt 1 2

3 4 Au 1

13

4 Re 1 1

3 4 Ai 1

23

4 Ap 1 3

3 4 Pd 1

43

3 Tc 1 4

3 2 Tr 1

43

1 Pn 1 4

3 1 Pn 1

43

2 Cb 1 4

3 3 1 3

2 1

4 2

2 1

4 3

2 2

4 3

2 3

4 2

2 3

4 1

2 2

4 1

2 2

4 1

2 3

4 1

2 3

4 2

2 2

4 3

2 1

4 3

2 1

4 2 2 4

411Hegm.Type 416AltruistType 415Altruist 414Revelation 413Alibi 412Asym Pd 111Prisoner'sD 116TotalConflict 115Tragedy 114Patron 113Patron 112Called Bluff 1

Lc 1 3

4 4 Al 1

24

4 Pl 1 1

4 4 Ac 1

14

4 Aa 1 2

4 4 Sh 1

34

4 Ap 1 4

4 3 Ha 1

44

2 Bb 1 4

4 1Mc 1

44

1 In 1 4

4 2 En 1

44

3 1 4

2 1

3 2

2 1

3 3

2 2

3 3

2 3

3 2

2 3

3 1

2 2

3 1

2 2

3 1

2 3

3 1

2 3

3 2

2 2

3 3

2 1

3 3

2 1

3 2 2 3

321LowConflict 326Aligned 325PureAligned 324Asym Coord 323Asym Assur 322Stag Hunt 221Asym Pd 226Hamlet 225Big Bully 224MissileCrisis 223Inspector 222Endless 2Mu 1

34

4 Pv 1 2

4 4 Pv 1

14

4 Po 1 1

4 4 Ar 1

24

4 Aa 1 3

4 4 Ai 1

44

3 Se 1 4

4 2 Se 1

44

1 Fs 1 4

4 1 Tc 1

44

2 Ic 1 4

4 3 1 4

3 1

2 2

3 1

2 3

3 2

2 3

3 3

2 2

3 3

2 1

3 2

2 1

3 2

2 1

3 3

2 1

3 3

2 2

3 2

2 3

3 1

2 3

3 1

2 2 3 2

331Mutual 336Privileged 335Privileged 334Pure Coord 333Assurance 332Asym Assur 231Alibi 236SecondBest 235SecondBest 234Fixed Sum 233TragicCycle 232InferiorCycle 3Mu 2

34

4 Pv 2 2

4 4 Pv 2

14

4 Co 2 1

4 4 Po 2

24

4 Ac 2 3

4 4 Re 2

44

3 Se 2 4

4 2 Se 2

44

1 Pc 2 4

4 1 Qc 2

44

2 Bc 2 4

4 3 2 4

3 1

1 2

3 1

1 3

3 2

1 3

3 3

1 2

3 3

1 1

3 2

1 1

3 2

1 1

3 3

1 1

3 3

1 2

3 2

1 3

3 1

1 3

3 1

1 2 3 1

341Mutual 346Privileged 345Privileged 344Coordination 343PureCoord 342Asym Coord 241Revelation 246SecondBest 245SecondBest 244PursuitCycle 243QuasiCycle 242BiasedCycle 4Al 3

34

4 Ah 3 2

4 4Mh 3

14

4 Pv 3 1

4 4 Pv 3

24

4 Pl 3 3

4 4 Au 3

44

3 Be 3 4

4 2 Su 3

44

1 Ge 3 4

4 1 Ge 3

44

2 QtQt 3 4

4 3 3 4

2 1

1 2

2 1

1 3

2 2

1 3

2 3

1 2

2 3

1 1

2 2

1 1

2 2

1 1

2 3

1 1

2 3

1 2

2 2

1 3

2 1

1 3

2 1

1 2 2 1

351Aligned 356Asym Hm 355Mixed Hmny 354Privileged 353Privileged 352PureAligned 251Altruist 256Benevolent 255Subsym Be 254Generous 253Generous 252QuasiAltruistType5Pl 3

34

4Hm 3 2

4 4 Ah 3

14

4 Pv 3 1

4 4 Pv 3

24

4 Al 3 3

4 4 AtAt 3

44

3 BtBt 3 4

4 2 Be 3

44

1 Ge 3 4

4 1 Ge 3

44

2 Sd 3 4

4 3 3 4

1 1

2 2

1 1

2 3

1 2

2 3

1 3

2 2

1 3

2 1

1 2

2 1

1 2

2 1

1 3

2 1

1 3

2 2

1 2

2 3

1 1

2 3

1 1

2 2 1 2

361PureAligned 366Harmony 365Asym Hm 364Privileged 363Privileged 362Aligned 261AltruistType 266Benev.Type 265Benevolent 264Generous 263Generous 262SamaritanD 6Nc 2

34

4 Pl 2 2

4 4 Al 2

14

4Mu 2 1

4 4Mu 2

24

4 Lc 2 3

4 4 HtHt 2

44

3 MtMt 2 4

4 2 Hs 2

44

1 Dh 2 4

4 1 Sn 2

44

2 Hg 2 4

4 3 2 4

1 1

3 2

1 1

3 3

1 2

3 3

1 3

3 2

1 3

3 1

1 2

3 1

1 2

3 1

1 3

3 1

1 3

3 2

1 2

3 3

1 1

3 3

1 1

3 2 1 3

311No Conflict 316PureAligned 315Aligned 314Mutual 313Mutual 312LowConflict 211Hegm.Type 216BlackmailTy. 215Hostage 214Delilah 213Samson 212Hegemony 13 "4 2 "4 1 "4 1""4 2 "4 3 "4 4 #3 4 #2 4 #1 4 #1 4 #2 4 #31 "2 1 "3 2 "3 3 #2 3 #1 2 #1 2 #1 3 #1 3 #2 2 "3 1 "3 1 "2

Game Numbers: Layer: Row: Column. Symmetric across SW-NE diagonal. Swapping Row and Column positions swaps indices, and Layer for 2&4 © CC-BY-SA ß 2011.05.25

To find a game: Make ordinal 4<3<2<1. Put column with Row's 4 right, row with Column's 4 up. Find layer by alignment of 4s; then intersection of Row&Column payoffs.see Robinson & Goforth 2005 The Topology of the 2x2 Games: A New Periodic Table www.cs.laurentian.ca/dgoforth/home.html www.BryanBruns.com/2x2chart.html

3 1 6 25 4 3 2 1 6 5 4 3 2

4,1

1,4 3,3

2,2

↓↑

↓↑

↓↑

↑↑

↑↑

↑↑

↓↓

↓↓

↓↓

↑↓

↑↓

↑↓

2. Structures in the Topology of 2X2 Games

a. Twelve Symmetric Games on the Diagonal b. Dominant Strategies & Nash Equilibria c. 4 Layers, 12 Payoff Patterns, 144 Games4 1 6 5 4 3 2 1 6 5 4 3 2 1 4 ↓

↓↓↓↓↓↑↓↑↓↑↓↑↑↑↑↑↑↓↑↓↑↓↑ 1 4 1 6 5 4 3 2 1 6 5 4 3 2 1

2 Chicken (Hawk-Dove) Ch ↑↓ 421 426 425 424 423 422 121 126 125 124 123 122 2 Hg Sd QtQt Bc Ic En Cb Bu UtUt Sb Ab Ch 2

3 Battle of the Sexes Ba ↑↓ 431 436 435 434 433 432 31 136 135 134 133 132 3 Sn Ge Ge Qc Tc In Pn Pr Pr Qb Ba Ab 3

4 Hero Hr ↑↓ 441 446 445 444 443 442 141 146 145 144 143 142 4 Dh Ge Ge Pc Fs Mc Pn Pr Pr Hr Qb Sb 4

5 Prisoner's Delight De (Anti-Ch) ↓↓ 451 456 455 454 453 452 151 156 155 154 153 152 5 Hs Be Su Se Se Bb Tr Ad De Pr Pr UtUt 5

6 Deadlock Dl (Anti-Pd) ↓↓ 461 466 465 464 463 462 161 166 165 164 163 162 6 MtMt BtBt Be Se Se Ha Tc Dl Ad Pr Pr Bu 6

1 PRISONER'S Pd DILEMMA ↓↓ 411 416 415 414 413 412 111 116 115 114 113 112 1 HtHt AtAt Au Re Ai Ap Pd Tc Tr Pn Pn Cb 1

2 Sh Stag Hunt ↓↑ 321 326 325 324 323 322 221 226 225 224 223 222 2 Lc Al Pl Ac Aa Sh Ap Ha Bb Mc In En 2

3 Ar Assurance ↓↑ 331 336 335 334 333 332 231 236 235 234 233 232 3 Mu Pv Pv Po Ar Aa Ai Se Se Fs Tc Ic 3

4 Co Coordination ↓↑ 341 346 345 344 343 343 241 246 245 244 243 242 4 Mu Pv Pv Co Po Ac Re Se Se Pc Qc Bc 4

5 Mh Mixed Harmony ↑↑ 351 356 355 354 353 35 251 256 255 254 253 252 5 Al Ah Mh Pv Pv Pl Au Be Su Ge Ge QtQt 5

6 Hm Harmony ↑↑ 361 366 365 364 363 362 261 266 265 264 263 262 6 Pl Hm Ah Pv Pv Al AtAt BtBt Be Ge Ge Sd 6

1 Nc No Conflict ↑↑ 311 316 315 314 313 312 211 216 215 214 213 212 1 Nc Pl Al Mu Mu Lc HtHt MtMt Hs Dh Sn Hg 1

3 34 24 14 14 24 34 43 42 41 41 42 43 Payoffs 3 Dominant strategies Nash Equilibria 2 3 212 13 23 32 31 21 21 31 32 23 13 12 2 D1 Column D0 None 1 0/2 4 games per tile, 36 games and 9 tiles per layer

Symmetric Quasi-symmetric Sub-symmetric D2 Double D1 Row 1 1 66 asymmetric pairs: 66 + 12 = 78 "unique" gamesd. High swaps (3!4) Link Tiles Across Layers 4 1 f. Interests Aligned, Mixed, or Opposedconnecting equivalently-located tiles on different layers 2 1 1 1 1 1 1 2 2 2 1 1 2 → → ↓ ↓ ← ← ↑ ↑ 2

6 Hotspots double-link 2 tiles # # N NE 3 2 1 1 2 2 2 1 1 1 1 1 1 ↓ ← ↑ → 3

6 Pipes link 4 tiles on 4 layers W # # E 4 2 1 1 2 3 2 1 1 1 1 1 1 ↓ ← ↑ → 4

1 3 Coord.-Battle Hotspot SW S # # 5 2 1 1 2 2 3 2 2 2 1 1 2 ↑ ↑ ← ← ↓ ↓ → → 5

NE Pd Pipe (Pd scrolled to northeast corner to unify tiles) 6 2 1 1 2 2 3 3 2 2 1 1 2 → ↓ ← ↓ 6

4 6 5 4 3 2 1 6 5 4 3 2 1 1 1 1 1 1 1 1 1 2 3 2 1 1 2 → ↓ ← ↓ 1

1 AtAt Au Re Ai Ap HtHt | Tc Tr Pn Pn Cb Pd 2 Lc Al Pl Ac Aa Sh 1 3 3 2 2 1 → → ↑ ↑ ← ← ↓ ↓ 2

2 Sd QtQt Bc Ic En Hg | Bu UtUt Sb Ab Ch Cb 3 Mu Pv Pv Po Ar Aa 1 2 2 3 2 1 ↑ → ↓ ← 3

3 Ge Ge Qc Tc In Sn X Pr Pr Qb Ba Ab Pn 4 Mu Pv Pv Co Po Ac 1 2 2 2 2 1 ↑ → ↓ ← 4

4 Ge Ge Pc Fs Mc Dh X Pr Pr Hr Qb Sb Pn 5 Al Ah Mh Pv Pv Pl 1 1 1 1 1 1 ↑ ↑ → → ↑ ↑ ← ← 5

5 Be Su Se Se Bb Hs — Ad De Pr Pr UtUt Tr 6 Pl Hm Ah Pv Pv Al 1 1 1 1 1 1 → ↑ ← ↑ 6

6 BtBt Be Se Se Ha MtMt — Dl Ad Pr Pr Bu Tc 1 Nc Pl Al Mu Mu Lc 1 2 2 2 2 1 → ↑ ← ↑ 1

| | X X — — | | X X — — 3 1 6 5 4 3 2 1 6 5 4 3 2 2 Externalities and inducement correspondences1 Pl Al Mu Mu Lc Nc | MtMt Hs Dh Sn Hg HtHt e. Remediability: Getting to Win-win Jekyll-Hyde TypeJekyll-Hyde Type ++ ↑ Pure Cooperation2 Al Pl Ac Aa Sh Lc | Ha Bb Mc In En Ap Swaps to reach win-win 1= single 3!4 swap ± ← →3 Pv Pv Po Ar Aa Mu X Se Se Fs Tc Ic Ai Paths with 2 or 3 steps include 2!3 and 1!2 swaps Pure Conflict - - ↓ Jekyll-Hyde TypeJekyll-Hyde Type4 Pv Pv Co Po Ac Mu X Se Se Pc Qc Bc Re Arrows show vectors for slices of games Fixed Rank-sum - - ± + +5 Ah Mh Pv Pv Pl Al — Be Su Ge Ge QtQt Au # Bold = each player has a pathway (Zero Sum) see Schelling 1963 The Strategy of Conflict6 Hm Ah Pv Pv Al Pl — BtBt Be Ge Ge Sd AtAt 2 Pareto-efficient paths: each swap step leaves 2 Greenberg 1990 The Theory of Social Situations

3 High swaps realign 4s; switch row or column in tile player with same or better-ranked outcome Robinson & Goforth 2005 The Topology of 2x2 Games

g. Games with Ties are Within the Topology h. Rapoport, Guyer & Gordon Taxonomy 4 i. Brams Typology and Game Numbers 16 5 4 3 2 1 6 5 4 3 2 1 2 55 50 49 70 78 72 39 35 36 65 67 66 50 37 36 46 31 29 22 18 19 52 53 57 2

1 3 56 52 51 74 76 71 37 31 32 64 68 67 56 39 38 43 45 47 20 14 15 51 54 53 3

2 4 44 41 40 73 75 77 38 33 34 69 64 65 49 13 12 42 44 30 21 16 17 55 51 52 4

3 5 18 16 15 53 42 45 10 8 7 34 32 36 6 4 3 40 23 25 10 8 7 17 15 19 5

4 6 17 14 13 54 43 46 11 9 8 33 31 35 5 2 1 41 24 26 11 9 8 16 14 18 6

5 1 21 19 20 57 47 48 12 11 10 38 37 39 35 33 34 48 27 28 32 11 10 21 20 22 1

6 2 26 22 23 58 62 61 48 46 45 77 71 72 Number of non-myopic 28 26 25 30 47 29 2

3 27 24 25 59 63 62 47 43 42 75 76 78 equilibria (NMEs) 27 24 23 44 45 31 3

1 4 30 28 29 60 59 58 57 54 53 73 74 70 3 2 1 48 41 40 42 43 46 4

2 5 2 4 5 29 25 23 20 13 15 40 51 49 All Nash equilibria are 34 1 3 12 38 36 5

3 6 1 3 4 28 24 22 19 14 16 41 52 50 NMEs except 33 2 4 13 39 37 6

4 1 6 1 2 30 27 26 21 17 18 44 56 55 pareto-deficient 35 5 6 49 56 50 1

5 Stable Weakly Stable Unstable 3 1 6 5 4 3 2 1 6 5 4 3 2 26 No Conflict N EP D2o D1o D0o D2c D1c D0c E=Natural outcome is Equilibrium Strongly cyclic

Strict Nonstrict:Half-swaps 2=3 1=2 Mixed M EP D2o D1o D2t D1t D1f D0f D1tf D1ctf 2 1 0 Dominant strategies_ Not cyclic Moderately cyclic

Ordinal games at grid intersections (graph nodes/vertices) Motive M Ep D2o pareto-deficient D1f D1tf no pressures No conflict Weakly cyclic

Games with ties (non-strict) lie between strict ordinal games, M e no D0 2 equilibria e2o e2c competitive pressure see Brams 1994 Theory of Moves; Rh 1

24

4 Pk 1 4

3 3 M e dominance D0 No equilibria e0o e0c threat-vulnerable Difficult see Brams & Kilgour 2008

as do all normalized 2 2

2 1

1 1

4 1 Complete Opposition Z E D2o D1o e eD0c force-vulnerable © CC BY-SA ß 2011.05.25

2x2 games (1-4 scale) Rousseau'sHuntPrisoner's Chkn see Rapoport et al.1976 2x2 Games; Robinson & Goforth 2003

see

Dra

gice

vic'

s G

ame

Theo

ry Ic

ons:

www

.lri.f

r/~dr

agice

/gam

eico

ns

(Robinson et al. 2007)

www.BryanBruns.com/2x2chart.html

23 14 32 14 31 24 21 34 12 34 13 24 14 23 14 32 24 31 34 21 34 12 24 13

NE

E

1 4

N

1 3

S

1 2

W

NE

E

1 4

N

2 4

3 4

W

SW

NE

E

2 3

N

2 4

S

1 2

SW

NE

E

2 3

N

1 3

S

3 4

W

SW

A Brief Overview of the Topology of 2x2 Ordinal Games The topology of 2x2 games (Robinson and Goforth 2005) elegantly displays the relationships between strict ordinal 2x2 games, where two players each have two strategies and four differently ranked preferences for the outcomes. Swaps in the two lowest payoffs (1!2) form tiles of four games closest to each other in the payoff space. Swaps in middle payoffs (2!3) and additional low swaps form layers. The four layers differ by the alignment of 4s. Each layer is a torus. Scrolling Prisoner's Dilemma (Game 111) to the center helps visualize the structure of the topology. The 12 symmetric games, where players face identical payoffs, form a diagonal axis from southwest to north-

east. Row payoffs are the same across rows, and column payoffs the same down columns, so the 66 asymmetric games on either side of this axis of symmetry combine payoffs from different symmetric games, for a total of 12 + 66 = 78 “unique” games.

Three rows in each layer have dominant strategies for Row (with higher payoffs whatever Column does), as do three columns for Column. Based on dominant strate-gies, three-fourths of games have a single Nash Equilib-rium, (a pair strategies that are best replies to each other). Battles of the Sexes and Stag Hunts, including asymmetric variants, have no dominant strategies and two equilibria. Cyclic games have no dominant strategies and (for ordinal payoffs) no equilibria.

Prisoner's Dilemma Family games have a Pareto-superior outcome that both players would prefer to the Nash Equilibrium. The family can extend to include Tragic games, also with a poor equilibrium but lacking a better alternative. In Second-Best games such as Pris-oner's Delight and Deadlock, both players can achieve their second-ranked preference (3,3). Biased games with high but unequal (4-3) equilibria form the largest payoff family. In Altruistic games, the largest subfamily, a player with a dominant strategy gets their second-ranked payoff. In Selfish games, the dominant strategy gets the best of a biased equilibrium. Unfair games have highly unequal (4-2) equilibria. Swaps in high payoffs (3!4) link layers. In six hot-spots, swaps for Row or Column connect the same two tiles, double-linking two layers, as in the Layer 1-3 Hot-spot where Battles of the Sexes turn into Coordination games. In six pipes, high swaps for Row or Column link to different tiles, weaving four tiles on four layers. High swaps, one for each player, convert Prisoner's Dilemma into an Asymmetric Prisoner's Dilemma (412 or 221) and then into the ordinal Stag Hunt (322). Most games, except fixed sum, can be converted to win-win through one or two swaps. Most games have mixed interests: a strategy may help or hurt other player. Games of pure cooperation or con-flict are less frequent, and fixed-sum (zero-sum) games even rarer. In Jekyll-Hyde Type games, such as Black-mail (216/461), incentives make one player kind and the other cruel. Games with ties lie between the strict ordinal games, linked by half-swaps that make (or break) ties in prefer-ences. For games with interval-scale or real payoffs, normalized versions can be mapped into the topology. Game numbers uniquely identify similar and related games, like scientific names for species, and so could aid comparative and cumulative research.

Harmony. Row or Column’s choice benefits the other, and leads to win-win

Prisoner’s Dilemma Tile: Swaps in lowest two payoffs trans-form Prisoner’s Dilemma into Chicken. Prisoner’s Dilemma: If both prisoners both keep silent, they each get a light sentence, their second-best outcome (3,3). If only one confesses, he goes free and the other gets a long sen-tence. But the dominant strategy for each is to confess (defect), ending up with both getting a medium-length sentence, the second-worst outcome (2,2). Chicken: If both play a Hawk strategy, both die (1,1). If both play Dove, both get second-best (3,3). Two Hawk-Dove Nash Equilibria yield very unequal outcomes (4,2 and 2,4).

Rousseau's Stag Hunt. A hunter could choose to cooperate in the risky hunt for a stag, or could catch a hare regardless of what the other hunter does. This game with ties lies between Assurance (333) and the strict Stag Hunt (322).

Samaritan’s Dilemma. Column could help him-self, without Row’s aid. Row prefers that both invest but prefers to help whatever Row does, while Column most prefers to be helped and do less.

Deadlock.Incentives lead to second-best

Battle of the Sexes. A couple prefers doing something together, but each has a different first choice

Fixed Rank-sum. Ordinal equivalent of Zero-sum; gains for one are matched by losses for the other. Cyclic, since one player always has an incentive to move. Maximin strategies avoid the worst payoff.

1 24 4

1 24 4

1 34 4

2 33 1

2 23 1

2 23 1

C Assurancesw Stag Hunt

1 24 4

1 24 4

1 34 4

2 32 1

2 22 1

2 22 1

Rousseau's Stag Hunt

1 24 4

1 24 4

1 34 4

3 32 1

3 22 1

3 22 1

Assurance RAssurancenw

Fs 3 21 4

2 34 1

443 Fixed Sum

Ba 3 42 2

1 1 4 3

133 Battle

Sd 3 44 3

1 1 2 2

262 SamaritanD

Hm3 2 4 4

1 1 2 3

366 Harmony

Cb 1 43 3 Pd 1 4

3 3

2 1 4 22 2 4 1

112Called Bluff 111PrisonersD

Ch2 43 3Cb 2 4

3 3

1 14 2 1 2 4 1

122 Chicken 121 Called Bluff

Dl 1 42 2

3 34 1

166 Deadlock