Periodic Table of 2x2 Games
Strict ordinal games
Symmetric games on diagonal axis
Left
Right
Payoffs
Up 1
4 3 3 Row Column
Nash equilbrium
Down 2
2 4 1 Pareto-inferior
Prisoner's Dilemma
L4 Nc
Ha
Pc
2 3 3 4 2 2 3 4 2 1 3 4
Ch 1 1 4 2 1 1 4 3 1 2 4 3
ChNc
Ba
Hr
ChHa
ChPc
3 3 2 4 3 2 2 4 3 1 2 4
1 1 4 2 1 1 4 3 1 2 4 3
BaNc
BaHa
BaPc
3 3 1 4 3 2 1 4 3 1 1 4
2 1 4 2 2 1 4 3 2 2 4 3
HrNc
HrHa
HrPc
2 3 1 4 2 2 1 4 2 1 1 4
Cm 3 1 4 2 3 1 4 3 3 2 4 3
CmNc
Dl
Pd
Sh
As
Co
Pc
Ha
Nc
CmHa
CmPc
1 3 2 4 1 2 2 4 1 1 2 4
3 1 4 2 3 1 4 3 3 2 4 3
DlNc
DlHa
DlPc
1 3 3 4 1 2 3 4 1 1 3 4
2 1 4 2 2 1 4 3 2 2 4 3
PdNc
PdHa
PdPc
Harmonious
Payoff swaps link neighboring games
Payoff Families
Stag Hunt
1↔2 Low swaps form tiles of 4 games
Win-win 4,4
Battle
2↔3 Middle swaps join tiles into 4 layers
Tile
Biased 4,3
Self-serving
Benevolent
Samaritan
3↔4 High swaps cross layers, bonding bands of tiles
CC9966
Scrolling Prisoner's Dilemma to center shows relationships
Second Best 3,3
Winner
Loser
Layers and table (toruses) wrap side-to-side & top-to-bottom
Unfair 4,2
Sad 3,2
Layers differ by alignment of best payoffs
Inferior
Dilemma 2,2
Alibi 3,2
Payoffs from symmetric games form asymmetric games
Indeterminate
High swaps turn Pd into Asym Dilemma (ShPd) and Stag Hunt
Cyclic
Co
As
Sh
Pd
Dl
Cm
Hr
Ba
Ch L1
2 1 3 4 2 2 3 4 2 3 3 4
2 4 3 3 2 4 3 2 2 4 3 1 2 4 3 1 2 4 3 2 2 4 3 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
ChCo
ChAs
ChSh
ChPd
ChDl
ChCm
ChHr
ChBa
Chicken
3 1 2 4 3 2 2 4 3 3 2 4
3 4 2 3 3 4 2 2 3 4 2 1 3 4 2 1 3 4 2 2 3 4 2 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
BaCo
BaAs
BaSh
BaPd
BaDl
BaCm
BaHr
Battle-Chicken
Battle
3 1 1 4 3 2 1 4 3 3 1 4
3 4 1 3 3 4 1 2 3 4 1 1 3 4 1 1 3 4 1 2 3 4 1 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
HrCo
HrAs
HrSh
HrPd
HrDl
HrCm
Quasi Battle
Hero-Chicken
Hero
2 1 1 4 2 2 1 4 2 3 1 4
2 4 1 3 2 4 1 2 2 4 1 1 2 4 1 1 2 4 1 2 2 4 1 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
CmCo
CmAs
CmSh
CmPd
CmDl
Unfair Type
Compromise Comp.-Hero Protector
1 1 2 4 1 2 2 4 1 3 2 4
1 4 2 3 1 4 2 2 1 4 2 1 1 4 2 1 1 4 2 2 1 4 2 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
DlCo
DlAs
DlSh
DlPd
Deadlock-Hero Deadlock-Battle Bully
Deadlock Asym Lock
1 1 3 4 1 2 3 4 1 3 3 4
1 4 3 3 1 4 3 2 1 4 3 1 1 4 3 1 1 4 3 2 1 4 3 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
PdCo
PdAs
PdSh
Dilemma-Hero Patron
Called Bluff
Prisoner D Total Conflict Misery
1 3 4 4 1 2 4 4 1 1 4 4 1 1 4 4 1 2 4 4 1 3 4 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
ShNc
ShHa
ShPc
ShCo
ShAs
Stag Hunt
1 3 4 4 1 2 4 4 1 1 4 4 1 1 4 4 1 2 4 4 1 3 4 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
AsNc
AsHa
AsPc
AsCo
Assurance Asym Assur.
2 3 4 4 2 2 4 4 2 1 4 4 2 1 4 4 2 2 4 4 2 3 4 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
CoNc
CoHa
CoPc
Coordination Pure Assur. Asym. Coord.
3 3 4 4 3 2 4 4 3 1 4 4 3 1 4 4 3 2 4 4 3 3 4 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
PcNc
PcHa
Peace-Coord. Privileged
Pure Aligned
Peace
3 3 4 4 3 2 4 4 3 1 4 4 3 1 4 4 3 2 4 4 3 3 4 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
HaNc
Harmony Asym Harmony Harmony-Coord.Harmony-Assur. Harmony-Hunt
2 3 4 4 2 2 4 4 2 1 4 4 2 1 4 4 2 2 4 4 2 3 4 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
Anticipation
Concord Pure Harmony Concord-Peace Concord-Coord. Mutualism
1 4 4 3 1 4 4 2 1 4 4 1 1 4 4 1 1 4 4 2 1 4 4 3
2 2 3 1 2 3 3 1 2 3 3 2 2 2 3 3 2 1 3 3 2 1 3 2
Asym Dilemma Hamlet
Big Bully
Crisis Cycle
Inspector Cycle Endless Cycle
1 4 4 3 1 4 4 2 1 4 4 1 1 4 4 1 1 4 4 2 1 4 4 3
3 2 2 1 3 3 2 1 3 3 2 2 3 2 2 3 3 1 2 3 3 1 2 2
Alibi
Assurance-Lock 2nd Best
ZeroSum Cycle Tragic Cycle
Inferior Cycle
2 4 4 3 2 4 4 2 2 4 4 1 2 4 4 1 2 4 4 2 2 4 4 3
3 2 1 1 3 3 1 1 3 3 1 2 3 2 1 3 3 1 1 3 3 1 1 2
Revelation
Coord.-Lock
Coord.-Compro. Pursuit Cycle
Quasi Cycle
Biased Cycle
3 4 4 3 3 4 4 2 3 4 4 1 3 4 4 1 3 4 4 2 3 4 4 3
2 2 1 1 2 3 1 1 2 3 1 2 2 2 1 3 2 1 1 3 2 1 1 2
Impure Altruist Peace-Lock
Benevolent
Peace-Hero
Generous
Samaritan Type
3 4 4 3 3 4 4 2 3 4 4 1 3 4 4 1 3 4 4 2 3 4 4 3
1 2 2 1 1 3 2 1 1 3 2 2 1 2 2 3 1 1 2 3 1 1 2 2
Altruist Type
BenevolentType Harm.-Compro. Harmony-Hero Harmony-Battle Samaritan's D
2 4 4 3 2 4 4 2 2 4 4 1 2 4 4 1 2 4 4 2 2 4 4 3
1 2 3 1 1 3 3 1 1 3 3 2 1 2 3 3 1 1 3 3 1 1 3 2
Hegemon Type Blackmail Type Hostage
Delilah
Samson
Hegemony
www.BryanBruns.com
Based on Robinson & Goforth 2005 The Topology of the 2x2 Games: A New Periodic Table www.cs.laurentian.ca/dgoforth/home.html
For more diagrams, explanations, and references, see Changing Games: An Atlas of Conflict and Cooperation in 2x2 Games www.2x2atlas.org
To find a game: Make ordinal 1<2<3<4. Put column with Row's 4 right row with Column's 4 up; find layer by alignment of 4s; find symmetric games with Row & Column payoffs.
L3 CC-BY-SA 2014.14.02
Symmetric Games with Ties
Low Ties
ln
lh
lo
ld
Games with ties lie between strict ordinal games, linked by half-swaps that make or break ties. For example,
Low Battle is between Battle and Hero, and Middle Battle (Volunteer's Dilemma) is between Chicken and Battle
Middle Ties
lk
lb
mh
Low Battle
Mid Harmony Mid Peace
1 3 4 4 3 1 4 4 1 1 4 4 1 4 3 3 1 4 1 1 3 4 1 1
1 1 3 1 1 1 1 3 3 3 1 1 1 1 4 1 3 3 4 1 1 1 4 3
Low Concord Low Harmony Low Coordination
Low Dilemma Low Lock
High Ties
hn
2 4 4 4 2 4 4 4
1 1 4 2 1 1 4 2
High Concord =High Chicken
Zero
ze
0 0 0 0
0 0 0 0
Zero
mp
mu
mk
mm
Basic
bh
hh
hm
4 2 4 4 2 4 1 1
1 1 2 4 4 4 4 2
High Harmony ≈HiCompromise
bd
Mid Hunt
Midlock
MidCompromiseMid Battle
1 1 4 4 1 4 1 1
1 1 1 1 1 1 4 1
Basic Harmony Basic Dilemma
hp
hk
ho
hs
hu
hd
he
hb
High Peace
≈High Lock
High Coord.
≈High Assurance
High Hunt
=High Dilemma
High Hero
≈High Battle
do
de
du
dn
Double Hunt
=DoubleConcord
4 1 4 4 1 4 2 2
2 2 1 4 4 4 4 1
Triple Ties
th
mb
3 3 4 4 3 1 4 4 1 3 4 4 1 4 3 3 3 4 1 1 3 4 3 3
1 1 3 3 3 3 1 3 3 3 3 1 3 3 4 1 3 3 4 3 1 1 4 3
Making high ties (and double ties) creates duplicate games, identical or equivalent by switching rows or columns
hc
L2
tk
4 4 4 4 4 1 4 4
1 1 4 4 4 4 1 4
Triple Harmony Triple Lock
2 1 4 4 1 2 4 4
4 4 1 2 4 4 2 1
Double Ties
dh
dp
4 1 4 4 4 1 4 4
1 1 1 4 1 1 1 4
DoubleHarmony =Double Peace
1 4 4 4 1 4 4 4
2 2 4 1 2 2 4 1
1 1 4 4 4 4 1 1
4 4 1 1 1 1 4 4
Double Coord. ≈Double Hero
4 4 1 1 4 4 2 2
2 2 4 4 1 1 4 4
1 4 4 4 1 4 4 4
1 1 4 1 1 1 4 1
Transforming Conflict and Cooperation in Strategic
Situations
A map to win-win, a menu of models, a chart for changing games
a. Twelve payoff patterns make 144 games
b. Dominant strategies define quadrants
The Robinson-Goforth topology of payoff
4
1
swaps conveniently arranges two-person twoCh
⇅
Ch
move (2x2) games in a natural order.
Ba
⇅
Ba
Symmetric games on the diagonal. Games
Hr
Samaritan
Cyclic
Patron-Client
Battle
⇅
Hr
where each faces the same payoff pattern form
Cm
⇊
Cm
a diagonal axis. Payoff patterns from
Dl
⇊
Dl
symmetric games combine to make
Pd
Patron-Client
⇊ Jeky-ll-Hyde Type Second-Best Deadlock
Pd
asymmetric games, and so can give names for
Sh
Sh
games.
As
As
Payoff swaps link games. Starting from any
Co
Privileged
Stag
Hunt
Second-Best
Cyclic
Co
strict ordinal 2x2 game (with four ranked
Pc
⇈
Pc
payoffs and no ties), swaps in the lowest
Ha
⇈
Ha
payoffs (1↔2) generate a tile of four games.
Nc
⇈ Harmonious Privileged
Nc
Jekyll-Hyde type Samaritan
Middle swaps (2↔3) start new tiles. More low
Nc
Ha
Pc
Co
As
Sh
Pd
Dl
Cm
Hr
Ba
Ch
3
2
and middle swaps complete a layer of nine
Reflections around axis switch row & column positions
1 0/2 (if ordinal)
D1 Column D0 None
tiles and 36 games, forming a torus that wraps
4 games per tile, 36 games and 9 tiles per layer
1 1
D2 Both
D1 Row
top to bottom, and left to right. Swapping the
66 asymmetric pairs: 66 + 12 = 78 "unique" 2x2 games
Nash Equilibria
Dominant strategies
highest payoffs (3↔4) starts a new layer.
c. Payoff changes can turn games into win-win
d. Incentives induce externalities
Layers differ by the alignment of highest
Ch
1
1 1
1
1
1
2
2
2
1
1
2 Ch
payoffs. Scrolling Pd to the center splits tiles
Ba
2
1
1
2
2
2
1
1
1
1 1
1 Ba
so the entire chart is also a torus: high swaps
3
Hr
2
1
1
2
2
1
1
1
1 1
1 Hr
(3↔4) link layers, wrapping top to bottom and
3
Cp
2
1
1
2 2
2
2 2
1
1
2 Cm
left to right.
3
Dl
2
1
1
2 2
2
2 2
1
1
2 Dl
Dominant strategies define quadrants. In each
3
Pd
1
1 1
1
1
1
2
2
1
1
2 Pd
layer's lower left quadrant, both players have a
Sh
Sh 1 2 3 2 2 1 Sh
dominant strategy, better whatever the other
3
As
1
2 2
2
1 As
As
does, leading to a Nash Equilibrium, from
Co
1
2 2
2
2
1 Co
Co
which neither can unilaterally improve their
Pc
1
1
1
1
1
1 Pc
Pc
payoff. In neighboring quadrants, one has a
Ha
1
1
1
1
1
1 Ha
Ha
dominant strategy, so the other's best choice
Nc
1
2
2
2
2
1 Nc
Nc
leads to an equilibrium. In the upper right
Interests may be aligned, opposed, or mixed
Nc Ha Pc Co As Sh Pd Dl Cm Hr Ba Ch
quadrant, neither has a dominant strategy; with
Jekyll-Hyde Type
Pure Cooperation
Swaps
realign incentives to make win-win situations
no equilibrium in pure strategies,
as incancyclic
1=
single
3↔4
swap
reaches
win-win.
2
or
3
step
games; or two equilibria, as in Stag Hunts and
paths may include 2↔3 and 1↔2 swaps
Jekyll-Hyde Type
Pure Conflict
Battles in the coordination quadrants.
Zero Sum
# Bold = paths for both. Pareto-efficient paths, each
An elegant array. Social dilemmas, including
swap step results in same or better-ranked payoff
see Schelling 1963 The Strategy of Conflict
Prisoner's Dilemma, Chicken, Battles, and Stag
Zero-sum games are hardest to remedy
Greenberg 1990 Theory of Social Situations, R&G 2005
Hunts, form a compact connected region in the
e. High swaps (3↔4) link layers
f. Simpler games with ties form borderlines
space of 2x2 games. Most games can be
Pd
scrolled
to
upper
right
to
unify
tiles
Nc Ha Pc Co As Sh Pd Dl Cm Hr Ba Ch
transformed into win-win by a single swap.
Ch Ch
Equivalently-located bands of 3 tiles slide and spin to link
Games of pure conflict, where one person's
Ba
Ba
incentives always encourage moves that make C bands criss-cross diagonally. H bands slide in row or column.
Hr
Hr
6
Hotspots
double-link
pairs
of
tiles
on
two
layers
e.g.
1:3
things worse for the other, negative
Cm
Cm
6
Pipes
link
quartets
of
tiles
on
four
layers,
e.g.,
H::H
(Harmony)
Pipe
externalities, lie on a diagonal linking the
C
D
H
C
D
Dl
Dl
4 H
1
cyclic tiles, including the zero-sum (fixed
Pd
Pd
Pd
rank-sum) games. Zero-sum games are farthest D 3:4 D::C D::D
1:2 D::C D::D
Sh
Sh
Ch
from win-win. Most games have mixed
As
Ba
As
interests.
C
C::H 2:4 C::D
C::H 1:3 C::D
Co
Hr
Co
Mapping payoff space. If payoffs occur
Cm
Pc
Pc
randomly, then the chart shows the expected
H
H::H H::C 1:4
H::H H::C 1:4
Dl
Ha
Ha
proportions of different games. Half swaps
Nc
Nc
make games with ties, between strict games.
Nc
Ordinal
games
with
ties
(indifference)
Games with payoffs normalized to a 1-4 scale
D
1:2 D::C D::D
3:4 D::C D::D
Sh games, at grid intersections
lie between strict
map onto the topology, so the chart shows the
As Strict
Half-swaps: 2≈1
2≈3
3≈4
payoff space of all normalized 2x2 games, and C C::H 1:3 C::D
C::H 2:4 C::D
see Robinson, Goforth
and Cargill 2007.
Co
the adjacent possible of changes in payoffs.
Pc Normalizing payoffs mapa all 2x2 games onto the chart
Sources See Robinson and Goforth 2005 The
H
H::H H::C 2:3
H::H H::C 2:3
0
-1
3
4
4
3
Ha
-5 -1
Topology of 2x2 Games: A New Periodic Table
1
4
3
3
-3 -5
3
1
1
3
mu
mb
Dl Cm Hr Ba Ch Pd 2
and Bruns 2014 Changing Games: An Atlas of 3 Ha Pc Co As Sh Nc
-3
0
3
3
1
4
Rousseau's Hunt
Volunteer's Dilemma
Axelrod's Pd
Row swaps switch row, column swaps switch column
Conflict and Cooperation in 2x2 Games.