2x2 Taxonomy
by
John Bryant*
Rice University
© 2013
The analysis commences with specifications of the related conundrums Prisoner’s
Dilemma, Chicken, and Stag Hunt, in that most spare and interesting class of games, the
completely ordered 2x2 ordinal games. Exploiting the structure that these great conundrums
provide, a new 2x2 taxonomy is generated. Perhaps most notably this exercise suggests the
efficacy of game theoretic taxonomy: in particular herein in demonstrating the explanatory
power, and economy in information requirements, of simple dominance criteria, criteria
motivated by the Prisoner’s Dilemma, in revealing the sources and scope of indeterminacies and
instabilities emanating from Chicken and Stag Hunt, and in portraying the uniquely pivotal role
of these great conundrums.
Lucas Cranach the Elder,
Stag Hunt of Elector Friedrich III the Wise (1529)
[Photo in the Public Domain]
*John Bryant, Dept. of Economics-MS 22, Rice University, P. O. Box 1892, Houston, Texas
77251-1892, USA, johnbradburybryant@comcast.net, phone 713-348-3341, fax 713-348-5278
(kindly clearly mark “for John Bryant”)
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1. Introduction
Game Theory presents a fundamental problem, how to extend maximization to strategic
interaction. (Von Neumann and Morgenstern, c. 1944, Chapter I, 2. -2.2.4) This paper argues
that the simplest of dominance criteria do a remarkably good job of doing so. Where one might
reasonably expect a solution they prescribe it, and where an unambiguous solution is reasonably
viewed as problematic they do not prescribe one. Moreover, in prescribing solutions, they
economize on the information requirements for players; such economizing on information
requirements often being particularly important in crisis situations. The vehicle for making this
case on dominance criteria is a new taxonomy of the 2x2 ordinal games.
The structure of the new taxonomy is motivated by the three great conundrums of Game
Theory, the Prisoner’s Dilemma, Chicken and Stag Hunt. The Prisoner’s Dilemma, characterized
by a Nash equilibrium Pareto dominated by a non-equilibrium outcome, motivates the two
dominance criteria, one of which it meets, standard vector dominance, one of which it does not,
set dominance. Chicken and Stag Hunt meet neither dominance criterion, and are used to
structure the rest of the new taxonomy. Chicken is characterized by Pareto non-comparable
Nash equilibria, that is, conflict games, as famously studied by Thomas Schelling (1960, 1980),
while the Stag Hunt is characterized by Pareto ranked Nash equilibria. By “mixing” these four
categories of games, those exhibiting set dominance, those exhibiting vector dominance but not
set dominance, those exhibiting Pareto non-comparable Nash equilibria, and those exhibiting
Pareto ranked Nash equilibria, the rest of the taxonomy is filled out.
All three of these great conundrums have canonical two player two strategy
representations, thereby revealing the essence of the related problems they present. Needless to
say, two player two strategy (per player) games have played a major role in the development,
application and teaching of Game Theory, and in strategic analysis generally, and for good
reason. In many cases they do clearly reveal the essence of a problem. Further, these canonical
representations of the great conundrums belong to the class of completely ordered (no
indifference) 2x2 ordinal games, a fascinating class of games initially introduced by Rapoport
and Guyer in 1966. One advantage of this class is that it is small enough, 78 games, to permit a
manageable sized complete taxonomy, while including many interesting games. As game theory
has the “curse of dimensions” in the extreme, this is no small thing. Yet further, as properties
derived for these games follow from their strict rankings of outcomes, these properties also
characterize all versions of the games specifying utility functions implying the same strict
ordinal rankings of the outcomes. Thus, taking the complete taxonomy together, qualitative
properties are provided for a substantial class of two player two strategy games, all those not
exhibiting indifference. Hereafter, where it will not lead to confusion, “2x2 games” refers to
Rapoport and Guyer’s class of games.
Robinson and Goforth’s (2005) volume provides an interesting, and relatively intricate,
topological development of the structure of the 2x2 games as a class, which reflects the vitality
and variety of this area of research. This paper, nevertheless, stays with the standard diagram, or
payoff matrix, form of Rapoport and Guyer as it is “directly useful for analysing
2
behaviour.” (Robinson and Goforth, 2005, p. 25) Indeed it is the analysis of behavior that
motivates the development of the structure of the new, simple and intuitive, taxonomy of this
paper. Further, with their topological direction, Robinson and Goforth do not address many of
the issues confronted in this paper, including the role of the Stag Hunt in explaining collapse, the
particular pivotal role of the Prisoner’s Dilemma, Chicken and Stag Hunt, taken together, in the
structure of the 2x2 games, or set dominance and the economizing on information of the
dominance criteria. They do, however, emphasize the central role of the Prisoner’s Dilemma,
and appropriately so. Robinson and Goforth (2005) also includes a helpful glossary and
bibliography.
2. Prisoner’s Dilemma and Dominance
“Simplicity is the ultimate sophistication” --brochure for Apple II [and Da Vinci?] (Isaacson,
Steve Jobs, p. 80)
The most famous, or infamous, type of game is the Prisoner’s Dilemma, which can be
represented by the following canonical symmetric standard 2x2 game diagram.
Canonical Prisoner’s Dilemma
Here (2,2) is the unique (standard vector) non dominated outcome (and hence a unique
strict Nash equilibrium), corresponding to both playing strategy one, as 2>1 and 4>3, yet it is
Pareto dominated by the (3,3) outcome. Thus the well deserved preeminence of the Prisoner’s
Dilemma. Notice that here the player does not need to know the other player’s payoffs to
determine dominance, just which of the other’s strategies corresponds to which of her own
payoffs. So one might naturally think of this as a situation not involving strategic interaction.
3
But the predicted outcome of (2,2) being Pareto dominated suggests that this may be jumping to
a conclusion too quickly.
Consider then the following game, and compare it to the Prisoner’s Dilemma:
Canonical “Strategy Safe”
PRODUCE
DON'T PRODUCE
PRODUCE
(4,4)
(3,2)
DON'T PRODUCE
(2,3)
(1,1)
with suggestive names for strategies one and two added.
Here strategy one, “produce,” vector dominates strategy two, “don’t produce,” as 4>2 and
3>1 (for both players). But in strategy one, “produce,” also has the feature that the worst
outcome from strategy one, “produce,” 3, is better than the best outcome from strategy two,
“don’t produce,” 2. This is a case of set dominance, and the best possible outcome is predicted
by both of these forms of dominance criteria in this game. (Bryant, 1987) Notice that here the
player does not need to know anything about the other player’s payoffs to determine dominance.
While vector dominance itself is not a very heroic extension of maximization to strategic
interaction, set dominance is the least heroic such extension, and it imposes the least information
requirements. One might refer to this as a “strategy safe” situation. In the fog of economic
crises reduced access to information may play a substantial role. Depending only upon strict
ranking as they do, these dominance criteria are also the natural complements of Rapoport and
Guyer’s 2x2 ordinal games.
In the presentation of the new taxonomy of 2x2 games, all of those games exhibiting set
dominance, and all of those exhibiting vector dominance but not set dominance, like the
Prisoner’s Dilemma itself, are treated.
3. Multiple Nash Equilibria
4
As with set and vector dominance, strict pure strategy Nash equilibria are defined by
strict ordinal ranking, and they cannot be dominated. When there are more than one, as in
Chicken and Stag Hunt, the dominance criteria do not determine an outcome.
Chicken, games with multiple Pareto non-comparable Nash equilibria, that is, conflict
games, can be represented by the following canonical symmetric standard 2x2 game diagram :
Canonical Chicken
STRATEGY 1
STRATEGY 2
STRATEGY 1
(1,1)
(4,2)
STRATEGY 2
(2,4)
(3,3)
Like with the Prisoner’s Dilemma, each player would be best off playing strategy one when
the other plays strategy two. In the Prisoner’s Dilemma, however, this competitive nature of the
game is “masked” by strategy one being vector dominant for both players. In “Chicken” payoffs
1 and 2 are switched, and there is no prediction from the dominance criteria, all strategy pairs are
possible. Now, if a player is convinced that the other player will be “stubborn” and play strategy
one then that pessimistic player will play strategy two and receive his next to worst outcome, if
his supposition is correct, while if the other player is indeed “stubborn” and plays strategy one he
gets his best possible outcome. Thus there are two competing strict Nash equilibria. But if both
play “stubborn,” not knowing that the other will do so as well, both get their worst outcome of
all. The name “chicken” derives from a, truly stupid from the adult perspective, teenage “game”
where two drivers drive their cars at each other at high speed, and, if one swerves, that one is
“chicken,” cowardly. Clearly, to predict a particular outcome one would have to go beyond the
strategic structure itself as represented in the game.
In the presentation of the new taxonomy of 2x2 games, all of those games exhibiting two
Pareto non-comparable, competitive, Nash equilibria, like canonical Chicken itself, are treated.
5
“… post the 2008-09 crisis, the world economy is pregnant with multiple equilibria—selffulfilling outcomes of pessimism or optimism, with major macroeconomic implications.”
(Olivier Blanchard, Economic Counsellor and Chief Economist at the International Monetary
Fund, 2011, emphasis his)
The remaining of our conundrums is the Stag Hunt coordination game, games with
multiple, here two, Nash equilibria that are Pareto ranked. Stag Hunt can be represented by the
following canonical symmetric standard 2x2 game diagram :
Canonical Stag Hunt
STRATEGY 1
STRATEGY 2
STRATEGY 1
(4,4)
(1,3)
STRATEGY 2
(3,1)
(2,2)
This is a, small and structurally simple, “team” game, a game of strategic complementarity
as emphasized in Cooper (1999). As compared to the Prisoner’s Dilemma, the (2,2) Nash
equilibrium outcome is now Pareto dominated by a Nash equilibrium outcome, (4,4). Thus,
indeed, the best possible outcome for both is physically feasible, but is not predicted (or ruled
out) by the dominance criteria, which provide no prediction. This strategic structure captures the
notion of (joint) self-fulfilling prophesy: with strategy one being “produce” and strategy two
“don’t produce,” for both players if they believe the other will produce, they will produce, only if
they think (fear) the other will (may?) not produce, out of the same fear on his part, will they not
produce. This is, then, the essence of Blanchard’s (2011) “self-fulfilling outcomes of pessimism
or optimism.”
On the other hand, the (4,4) Nash equilibrium is Pareto optimal, and hence focal, and one
might naturally predict it to occur. However, experiments with the Stag Hunt coordination game
clearly demonstrate the likelihood of coordination-failure. In addition to coordination-failure,
the repeated play of these games can exhibit complicated dynamics, and there is not a consensus
6
on the theory of these dynamics. (Romer, 2012, p. 290). The appendix provides a brief
discussion of the particularly striking experiments of Van Huyck, Battalio and Beil (1990)
[VHBB] with Bryant’s (1983) large group, many strategy, game, which may be particularly
relevant to Macroeconomics. The experimental coordination game literature as a whole, in
which coordination-failure is widely observed, is now extensive, but a fine “entry point” is
Goeree and Holt, (2005) and Holt (c. 2007).
4. “Greed” and “Fear” in the Great Conundrums
With the three great conundrums in hand, further insight into their relationship can be
provided. Indeed, in the same venerable volume of General Systems as Rapoport and Guyer’s
paper, Morehous (1966) provides a valuable insight into the canonical representation of that
greatest conundrum, the Prisoner’s Dilemma, which insight can be extended to Chicken and Stag
Hunt. Take the “cooperative” outcome in the canonical Prisoner’s Dilemma, (3,3), to be the
norm. One problem with this norm is that a player who deviates from the norm when the other
does not is better off doing so. A second problem is that if the other player deviates from the
norm, a player is better off having deviated as well. Morehous suggestively labels these “greed”
and “fear.” Similarly, for the other two conundrums, in their canonical representations, take the
“cooperative” outcome to be the norm. In canonical Chicken this norm, (4,4), has the problem
that a player who deviates from the norm when the other does not is better off doing so, “greed.”
But it does not have the second problem. If the other player deviates from the norm, a player is
better off not having deviated. In contrast, in the canonical Stag Hunt this norm, (4,4), does not
have the first problem as a player who deviates from the norm, when the other does not deviate,
is worse off doing so. However, as described in (3) above, it does have the second problem, that
if the other player deviates from the norm, a player is better off having deviated as well, “fear.”
In the Stag Hunt experiments “fear” apparently often dominates the focal nature of the Pareto
optimal (4,4) Nash equilibrium outcome.
With these relationships between the great conundrums of Game Theory, the Prisoner’s
Dilemma, Chicken and Stag Hunt in hand, we are ready to turn to the new taxonomy in which
they play a pivotal role.
5. Taxonomy
To present the new taxonomy of 2x2 games there is some minor but necessary detail to go
through.
As mentioned in the Introduction (1) above, herein, where it will not lead to confusion,
“2x2 games” refers to the complete set of completely ordered (no indifference, or “strict”) 2x2
(two player two strategies each) ordinal games of Rapoport and Guyer. In 2x2 games the two
players can order from best to worst the four possible “outcomes,” with no cases of indifference,
but their evaluations can go no further than this. For expositional convenience these ordinal
payoffs are referred to as 4,3,2 or 1, with the understanding that the “outcome” receiving payoff
4 just means that it is strictly preferred to the “outcome” receiving payoff 3, and so on. In the
analysis itself, the strategies are only identified by their payoffs to the “outcomes,” they have no
independent identities of their own; and for, that matter, in the analysis itself, the payoffs are
attached to the cross product of strategies, “outcomes” is really just a suggestion of a sort of
7
situation to which the analysis applies.
In this context it is arbitrary whether a player’s strategy is labeled “strategy one” or
“strategy two.” (In application, however, sometimes there are symmetries in the actual
strategies available to the players, “run” or “stay” say, so that one may want to adjust for this in
using the taxonomy as presented.) As the numbering of the strategies is, at this level of
abstraction, arbitrary, to avoid duplication the convention is now adopted that 4 appears in the
first row of the row player’s payoffs, and in the first column of the column player’s payoffs.
These minor but necessary details help explain the structure of the “map” of the full
taxonomy, which is used for reference below. This taxonomy is generated by a structured
pairwise matching of the row and column player four element payoff matrices.
8
12
11
IxII
I
10
IxIII
IxIV
IIxIII
IIxIV
9
8
II
Row Axis
7
6
III
5
IIIxIV
4
3
IV
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
Column Axis
Roughly speaking, “I” refers to pure set dominance games, pairwise matching of payoff matrices
exhibiting set dominance, “II” to pure (standard) vector dominance games, both criteria
suggested by the Prisoner’s Dilemma, and “III” to pure Chicken games, and “IV” to pure Stag
Hunt games, as they, and the other categories, are described below. (The reader may also
observe below that “I” could be referred to as “IxI,” and so on.) With the taxonomy in hand,
then, for example, finding and characterizing those games in which it is physically possible for
both players to get the best possible outcome, and those games in which it is physically possible
for both to players get the worst possible outcome, is trivial: look for “(4,4)” and “(1,1)” games
respectively; and similarly for other common characteristics.
9
We turn to the first segment, or module, of the taxonomy. “I” is the pure set dominance
games, the set of games exhibiting set dominance for both players It is convenient to list just the
four payoffs to the column player in a qualifying game, leaving out the rest of the game’s
diagram. The possible set dominant payoffs for the column player are:
41
32
42
31
31
42
32
41
Now imagine the left side of the page has the respective row player payoffs running down it, in
the same order. Matching them up then results in a matrix of both players payoffs of the games.
Leaving out the rest of the diagram again, but this first time including the borders for clarity, we
have:
41
32
42
31
31
42
32
41
43
12
(4,4) (3,1)
(1,3) (2,2)
(4,4) (3,2)
(1,3) (2, 1)
(4,3) (3,1)
(1,4) (2,2)
(4,3) (3,2)
(1,4) (2,1)
43
21
canonical “Strategy Safe”
(4,4) (3,2)
(2,3) (1, 1)
(4,3) (3,1)
(2,4) (2,2)
(4,3) (3,2)
(2,4) (1,1)
(3,3) (4,1)
(1,4) (2,2)
(3,3) (4,2)
(1,4) (2,1)
34
12
34
21
(3,3) (4,2)
(2,4) (1,1)
(That’s ten games “down.”)
The payoffs below the principle diagonal are left out because of the symmetry in the row and
column players’ payoffs, and that the initial assignment of a player as row or column player is
arbitrary. For example, the missing payoffs in row three column two are the payoffs in row two
column three, but with row and column player reversed, while the games on the principle
diagonal are themselves symmetric, by construction.
All of these games are solvable by set (and vector) dominance. Here the player does not
need to know anything about the other player’s payoffs to determine dominance. For all of these
games in this class, call it Class I, the set dominance prediction is a payoff for each player of
either 4 or 3, depending on the game, including all combinations, and where (4,4) is physically
possible it is predicted.
In this context, of the least heroic extension of maximization to social interaction, it is
useful to first introduce a different information structure and related prediction mechanism,
contraction by set dominance. Suppose the column player, say, realizes that for the row player
strategy one is set dominant, and that the row player behaves accordingly and will choose
strategy one. Then to know which strategy to play the column player need only be able to rank
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her outcomes associated with the row player’s strategy one, and not with the row player’s
strategy two, for it is now, with this realization, a simple maximization problem. In further
reaches of the taxonomy, contraction by dominance (if feasible) can be the only applicable
mechanism of prediction by dominance.
“II” is the pure vector dominance games. This is a mercifully small class as the
possible payoffs for the column player, and respective game matrix are:
43
21
21
43
(4,4) (2,3)
(3,2) (1,1)
(4,2) (2,1)
(3,4) (1,3)
(2,2) (4,1) (canonical Prisoner’s Dilemma)
(1,4) (3,3)
(That’s thirteen games “down.”)
All of these games are solvable by vector dominance. The only game here, in this class,
exhibiting the problematic Pareto dominated, and yet unique non vector dominated outcome (and
hence Pareto dominated Nash equilibrium outcome) of the Prisoner’s Dilemma is the canonical
version of the Prisoner’s Dilemma already exhibited. Unlike set and vector dominance, Pareto
dominance depends upon preference orderings of both players and requires a correct matching.
In the asymmetric game, in the vector dominance prediction, one of the players, the one with the
Prisoner’s Dilemma payoff pattern itself, receives his next to worst outcome. Where it is
physically feasible, the best outcome for both is predicted by vector dominance and the worst
outcome for both eliminated. The vector dominant player does not need to know the other
player’s payoffs to determine dominance, just which of the other’s strategies corresponds to
which of her own payoffs.
In this vector dominance context it is useful once again to first introduce a different
information structure and the related prediction mechanism, contraction by vector dominance.
Suppose the column player, say, realizes that for the row player strategy one is vector dominant,
and that the row player behaves as prescribed by vector dominance, and will choose strategy one.
Then to know which strategy to play the column player need only be able to rank her outcomes
associated with the row player’s strategy one, and not with the row player’s strategy two, for it is
now, with this realization, once again, a simple maximization problem. In further reaches of the
taxonomy, contraction by vector dominance (if feasible) can be the only applicable mechanism
of prediction by dominance.
The remaining of our conundrums are the Chicken, and Stag Hunt coordination games,
that is games with multiple (here two) Nash equilibria. (Note that some use the term
“coordination game” for Stag Hunt games only.) As Nash equilibria cannot occur in the same
row or column, the pairs of Nash equilibria either have the players playing differently numbered
strategies, represented by Chicken, or both playing the same numbered strategies, strategy one or
11
strategy two, as in the Stag Hunt game. The associated two classes of games are those of the new
taxonomy that satisfy these conditions respectively.
“III” is the pure Chicken games. For games with conflicting Nash equilibria like
Chicken, Class III, the possible payoffs for the column player, and respective game matrix are:
23
41
12
43
13
42
(2, 2) (4, 3)
(3, 4) (1,1)
(2, 1) (4, 2)
(3, 4) (1,3)
(2, 1) (4, 3)
(3, 4) (1, 2)
(canonical Chicken) (1, 1) (4, 2)
(2, 4) (3, 3)
(1, 1) (4, 3)
(2, 4) (3, 2)
(1, 1) (4, 3)
(3, 4) (2, 2)
(That’s nineteen games “down.”)
All of these games, Class III, are conflict coordination games like canonical Chicken,
and, hence, dominance criteria provide no predictions. For three of them both playing
“stubborn” yields the worst possible outcome, with the symmetric canonical Chicken itself
having the largest “gap” between both playing stubborn and both playing “conciliatory.” In one
of the games, the first, upper left, symmetric one, both playing “stubborn” Pareto dominates both
playing “conciliatory.” In the remaining two games both playing “stubborn” and both playing
“conciliatory” are Pareto dominated. So not only are these not zero sum games, which is
meaningless in the ordinal context, they also have elements of common interest; as, indeed,
Schelling argues is typical in general in situations involving conflicts of interests. (Schelling,
1960, 1980, Chapter I) Here a best possible for both outcome is not physically feasible.
12
“IV” is the pure Stag Hunt games. For games with Pareto ranked Nash equilibria, Class
IV, the possible payoffs for the column player, and respective game matrix are:
41
23
42
13
43
12
(4, 4) (2, 1)
(1, 2) (3, 3 )
(4, 4) (2,2)
(1, 1) (3,3)
(4, 4) (2, 3)
(1, 1) (3, 2)
(4, 4) (1, 2)
(2, 1) (3, 3)
(4, 4) (1, 3)
(2, 1) (3, 2)
(canonical Stag Hunt) (4, 4) (1, 3)
(3, 1) (2, 2)
(That’s twenty-five games “down.”)
None of these games are solved by dominance. Indeed, all of these games, Class IV, have
Pareto ranked Nash equilibria like the canonical Stag Hunt, with the canonical Stag Hunt having
the largest gap between the good and bad Nash equilibrium. All Class IV games, then, provide
models of occasional collapse induced by switches between equilibrium beliefs. They are all
Stag Hunt games. In all of them the best possible outcome for both is physically feasible, but
not, of course, predicted by dominance, and where the worst possible outcome for both is
physically feasible, it is not ruled out of course. Two of the games have both players getting the
best possible outcome and both players getting the worst possible outcome, both situations, being
feasible and not ruled out.
With these classes of games, motivated by the Prisoner’s Dilemma and multiple Nash
equilibria, the basic structure of the taxonomy is determined. That is, the remainder of the
taxonomy is generated by “mixing” these four “Pure” classes. That is, we pairwise match row
and column players from the different classes. In doing so, the advantage of the parsimonious
2x2 structure becomes abundantly clear. Game Theory has the “curse of dimensions,” or
richness, in the extreme.
13
“IxII” is Class I x Class II of Set Dominance x Vector Dominance games.
The Class I row player payoffs are: 4 3
43
34
3 4 and
12
21
12
21
the Class II column player payoffs are:
43
21
21
43
and the resultant game matrix is:
(4, 4) (3, 3)
(1, 2) (2, 1)
(4, 2) (3, 1)
(1, 4) (2, 3)
(4, 4) (3, 3)
(2, 2) (1, 1)
(4, 2) (3, 1)
(2, 4) (1, 3)
(3, 4) (4, 3)
(1, 2) (2, 1)
(3, 2) (4, 1)
(1, 4) (2, 3)
(3, 4) (4, 3)
(3, 2) (4, 1)
(2, 2) (1, 1)
(2, 4) (1, 3)
(That’s thirty-three games “down.”)
As all of the “mixed” games are inherently asymmetric, these sub matrices are rectangular.
All of these “IxII” games are solved by dominance, with the upper left respective payoffs
being the outcome. But here, once again, there are two ways to generate these solutions, which
have different minimum requirements on the players’ information. The row player, with strategy
one set dominant, only needs to know that strategy one guarantees one of her two best outcomes.
In one case the vector dominant column player needs to know her own payoffs and how they
“line up” with the row player’s strategies (and be guided by the weaker vector dominance
criterion). Alternatively, if the column player knows that strategy one is set dominant for the row
player, and that the row player knows this and acts accordingly, and she knows her payoffs from
her strategies when the row player plays strategy one, then her’s is just a simple maximization
problem. This alternative information requirement provides a simple example of the, earlier
introduced, contraction by dominance (here set, and hence also vector dominance), an important
approach to many games (although with multiple strategies it need not yield a single predicted
outcome as here). Naturally, when the best outcome for both is physically possible it occurs, but
a mediocre predicted outcome of (3, 2) is possible, and the vector dominant column player can
do better (in ordinal terms that is) than the set dominant row player; (3, 4) can be the predicted
outcome.
14
“IxIII” is Class I x Class III of Set Dominance x Chicken games.
The Class I row player payoffs are: 4 3
43
34
3 4 again and
12
21
12
21
the Class II column player payoffs are:
23
41
12
43
13
42
and the resultant game matrix is:
(4, 2) (3, 3)
(1, 4) (2, 1)
(4, 1) (3, 2)
(1, 4) (2, 3)
(4, 1) (3, 3)
(1, 4) (2, 2)
(4, 2) (3, 3)
(2, 4) (1, 1)
(4, 1) (3, 2)
(2, 4) (1, 3)
(4, 1) (3, 3)
(2, 4) (1, 2)
(3, 2) (4, 3)
(1, 4) (2, 1)
(3, 1) (4, 2)
(1, 4) (2, 3)
(3, 1) (4, 3)
(1, 4) (2, 2)
(3, 2) (4, 3)
(3, 1) (4, 2)
(3, 1) (4, 3)
(2, 4) (1, 1)
(2, 4) (1, 3)
(2, 4) (1, 2)
(That’s forty-five games “down.”)
All of these are solvable by contraction by set dominance, but the “competitive” column
player never does better than the set dominant row player, and can do as poorly as 2. Here the
best outcome for both is physically infeasible but where the worst outcome for both is physically
feasible it is predicted not to occur.
15
“IxIV” is Class I x Class IV of Set Dominance x Stag Hunt games.
The Class I row player payoffs are: 4 3
12
43
21
The Class IV column player payoffs are now:
34
12
3 4 again and
21
41
23
42
13
43
12
and the resultant game matrix is:
(4, 4) (3, 1)
(1, 2) (2, 3)
(4, 4) (3, 2)
(1, 1) (2, 3)
(4, 4) (3, 3)
(1, 1) (2, 2)
(4, 4) (3, 1)
(2, 2) (1, 3)
(4, 4) (3, 2)
(2, 1) (1, 3)
(4, 4) (3, 3)
(2, 1) (1, 2)
(3, 4) (4, 1)
(1, 2) (2, 3)
(3, 4) (4, 2)
(1, 1) (2, 3)
(3, 4) (4, 3)
(1, 1) (2, 2)
(3, 4) (4, 1)
(3, 4) (4, 2)
(3, 4) (4, 3)
(2, 2) (1, 3)
(2, 1) (1, 3)
(2, 1) (1, 2)
(That’s fifty-seven games “down.”)
All of these are solvable by contraction by set dominance, and all the predicted outcomes
are good, 3 or 4.
Interestingly, the stag hunt column player always receives 4, and thus sometimes does
better, in his ordinal ranking, than does the set dominant row player. Parenthetically, one might
think that this would warrant the attention of analysts who believe that utility is endogenous, or
passed on somehow. However, a “converted” Stag Hunt player’s best outcome may be worse for
him than the second best outcome he could have gotten in his original set dominant state. The
rankings can be situation specific ordinal rankings. While behavior is altered, we do not in
general know whether or not there is an associated utility level that falls with the alteration.
Where the best outcome for both is physically possible it occurs.
Remembering the problematic aspect of the canonical Prisoner’s Dilemma (one of the
Class II games), the plot now thickens.
16
“IIxIII” is Class II x Class III of Vector Dominance (only) x Chicken games, an ominous,
problematic class in its own right!
The Class II row player payoffs are: 4 2
2 4 and
31
13
the Class III column player payoffs are:
23
41
12
43
13
42
and the resultant game matrix is:
(4, 2) (2, 3)
(3, 4) (1, 1)
(4, 1) (2, 2)
(3, 4) (1, 3)
(4, 1) (2, 3)
(3, 4) (1, 2)
(2, 2) (4, 3)
(2, 1) (4, 2)
(2, 1) (4, 3)
(1, 4) (3, 1)
(1, 4) (3, 3)
(1, 4) (3, 2)
(That’s sixty-three games “down.”)
All of these are solvable by contraction by vector dominance, with some poor outcomes
predicted, that is, some 2’s predicted, including 2’s for both. Strikingly, in the whole first row a
Pareto dominated outcome is predicted, as in the canonical Prisoner’s Dilemma itself. In these
Prisoner’s Dilemma games the vector dominant player exhibits both Morehous’s “Greed” and
“Fear” problems, as in the, symmetric, canonical Prisoner’s Dilemma. The Prisoner’s Dilemma
is relatively flourishing in this asymmetric case. Here the best outcome for both is never
physically possible.
“IIxIV” is Class II x Class IV of Vector Dominance (only) x Stag Hunt games.
The Class II row player payoffs are: 4 2
31
2 4 and
13
the Class IV column player payoffs are:
41
23
42
13
43
12
and the resultant game matrix is:
(4, 4) (2, 1)
(3, 2) (1, 3 )
(4, 4) (2,2)
(3, 1) (1,3)
(4, 4) (2, 3)
(3, 1) (1, 2)
(2, 4) (4, 1)
(2, 4) (4, 2)
(2, 4) (4, 3)
(1, 2) (3, 3 )
(1, 1) (3, 3)
(1, 1) (3, 2)
(That’s sixty-nine games “down.”)
All of these are solvable by contraction by vector dominance, with some poor outcomes
predicted, that is in the second row 2’s are predicted for the vector dominant player. But, once
again, the stag hunt player receives only 4’s, and, thus no Pareto dominated outcomes are
predicted. But when the best outcome for both is physically feasible it is predicted, and when the
worst outcome for both is physically feasible it is predicted not to occur.
17
Finally we have the intriguing “IIIxIV” which is Class III x Class IV or Chicken x Stag
Hunt.
The Class III row player payoffs are: 2 4
31
the Class IV column player payoffs are:
14
23
41
23
1 4 and
32
42
43
13
12
and the resultant game matrix is:
(2, 4) (4, 1)
(3, 2) (1, 3)
(2, 4) (4, 2)
(3, 1) (1, 3)
(2, 4) (4, 3)
(3, 1) (1, 2)
(1, 4) (4, 1)
(2, 2) (3, 3)
(1, 4) (4, 2)
(2, 1) (3, 3)
(1, 4) (4, 3)
(2, 1) (3, 2)
(1, 4) (4, 1)
(1, 4) (4, 2)
(1, 4) (4, 3)
(3, 2) (2, 3)
(3, 1) (2, 3)
(3, 1) (2, 2)
(Now that’s all seventy-eight games “down.”)
Dominance makes no prediction in any of these games. Further, all nine of these games
do not have any Nash equilibria at all. And these are the only of our games with no Nash
equilibria. This is, then, a proof by inspection that multiple Nash equilibria is not a necessary
condition for dominance criteria to fail to predict an outcome, perhaps an obvious theorem, but it
also demonstrates that this stems from mixing payoffs associated with the two opposing types of
multiple Nash equilibria. The payoffs are “pulling” for pairs of Nash equilibria in opposite
directions, as it were. The best outcome for both and the worst outcome for both are physically
infeasible in these games.
Having described all the classes of 2x2 games in the new taxonomy, a summing up of this
taxonomy’s features is called for.
6. Taxonomy: A Summing Up
How, then, does the new taxonomy present a significantly different perspective? In broad
strokes:
(A) The three great conundrums of game theory, the Prisoner’s Dilemma, Chicken and
Stag Hunt, play a unique pivotal role in the structure of the new taxonomy. Given the prominent
role that two player two strategy games have played in the development and teaching of Game
Theory, this is a significant insight.
(B) All games involving set dominance are determinate using set dominance or
contraction by set dominance. That is classes (I), (IxII), (IxIII), (IxIV).
(C) All games involving vector dominance but not set dominance are determinate using
vector dominance or contraction by vector dominance. That is classes (II), (IIxIII), (IIxIV).
(D) Both set and vector dominance economize on information requirements, with set
dominance economizing the most.
18
(E) Despite dominance criteria economizing on information requirements, all the
remaining games are recognized conundrums of Game Theory, involving either two Nash
equilibria, Chicken and Stag Hunt games, or no Nash equilibria.
(F) The pairwise “mixing” of Chicken and Stag Hunt games produces all of the games
without a Nash equilibrium, and only such games. That is class (IIIxIV).
(G) The four “pure” classes of games, set dominant, vector but not set dominant, Chicken
and Stag Hunt, and their pairwise “mixing” exhaust the 2x2 games.
But taxonomies may simply have value in their own right.
19
APPENDIX
VHBB and Strategic Uncertainty
“Van Huyck, Battalio and Beil’s results [VHBB] suggest that predictions from deductive theories
of behavior should be treated with caution: even though Bryant’s [1983] game is fairly simple,
actual behavior does not correspond well with the predictions of any standard theory. The results
also suggest that coordination-failure models can give rise to complicated behavior and
dynamics.” (Romer, David, Advanced Macroeconomics, 2012, p. 290)
We now face stark and perplexing results. Fortunately, the discussion of Rapoport and
Guyer’s 2x2 approach provides insight into the complex behavior and dynamics revealed in
VHBB’s repeated game experiments. These complexities themselves require the multiple player,
multiple strategies per player structure of their large Stag Hunt game however, a game motivated
by Macroeconomics, as noted by Romer (2012, p. 290) and by VHBB (1990, p. 235) themselves.
The striking, robust and widely replicated “downward cascades” or “self-fulfilling [and
intensifying] panics” observed in the repeated game experiments of John Van Huyck, Raymond
Battalio and Richard Beil in “Tacit Coordination Games, Strategic Uncertainty, and Coordination
Failure,” (1990) [VHBB] are experimental results that, I think it is fair to say, are now
considered classic, at least in their original Macroeconomics context. Moreover they may
ultimately have importance for issues not touched upon herein, or even imagined. For Game
Theory of itself is no modest enterprise, involving as it does all interaction between two or more
people; if not, indeed, involving more, for example competing portions of the nervous system
with competing access to sensory input and competing control of the body, or involving or
treating non human entities or organizations as actors. That is, it involves interactive systems
generally, it being no accident that Rapoport and Guyer (1966) and Morehous (1966) published
in a journal named General Systems.
Indeed, with respect to this lack of modesty, in its original formulation Game Theory
invoked a very powerful, but also very strong, in the sense of stringent, assumption. Namely, the
“one-shot” game characterization provides a complete description of the relevant environment.
Furthermore, therefore, in every environment that has the same “one-shot” game characterization
the same behavior occurs. The VHBB repeated game experiments provide one particularly stark
example where this assumption is most unambiguously violated. The perplexingly complex
behavior and dynamics generated in these experiments is a reflection of this stark violation of the
complete description assumption.
Needless to say, the violation of complete description has received a great deal of
attention in this context, as well as others, and with varying degrees of success in the resulting
analysis. (Holt, 2007, Chapter 26) So what may be most perplexing is that the dynamics in
VHBB has remained perplexing! (Romer, 2012, p. 290) VHBB’s strategic uncertainty has
proven refractory.
The final objective of the paper, then, is to provide qualitative insights into the perplexing
complicated behavior and dynamics of the VHBB experiments based on the following structure.
20
Faithfulness and Trust
are important, and in some situations critical:
“To focus the analysis consider the following tacit coordination game, which is a strategic form
representation of John Bryant’s (1983) Keynesian coordination game.” (VHBB, p. 235)
U(i) = A { minimum [e(1), ...,e(N)] } - B {e(i)},
i = 1, ...,N; A > B> 0; 0 ≤ e(i) ≤ E.
(Symmetric, Summary Statistic)
VHBB’s specification is of a multiple player multiple strategy team game. Numerous
individual’s efforts are complements. In the strategic form representation there are N>1
individuals, labeled as individual 1 through N, and individual i is one particular one of those N
individuals. Each individual i chooses an “effort level” e(i) that is between 0 and E, E>0. The
payoff to individual i is called U(i) and is increasing in the minimum effort level chosen in the
whole group of N individuals, and decreasing, but less strongly, in the individual’s own effort.
Intuitively, then, this can be thought of as similar to a simple model of the individual payoffs to
members of a “team.”
More technically, it is worth noting that if you are going to model team production, where
“moving together” beats “going it alone,” and if you are going to assume constant returns to
scale and also want the technical convenience of continuity in “effort” (hence the continuum of
equilibria) then, by Euler’s theorem, the payoff function must be non-differentiable, as with the
“min rule:” that is, the no surplus condition (a condition of Walrasian equilibrium not met in
team production) is not a mere technicality, consider also the more familiar Leontief production
technology specification. Notice further that as the number of players grows the “min rule”
provides a “severe test of payoff dominance” (VHBB, p. 236), a feature completely missed by
standard theory.
As for experimental suitability, the equality of some payoffs, but with the Nash equilibria
still Pareto ranked, the symmetry and the summary statistic (“min rule” here), these features
make possible the compact and clear formulation, which clarity is important for experiments. In
addition, as a practical matter, the symmetry means one observes that many more individuals
facing the same problem.
In their experimental form VHBB used seven levels of “effort,” and payoffs were in
dollars. In the first experiment there were seven groups of 14-16 subjects playing ten repeated
games with the same group. In the first play almost 70% did not play the high, best for all, if all
do so, effort level, but only 2% played the lowest effort level. Only 10% predicted there would
be an equilibrium outcome, and none of the seven groups played an equilibrium. In subsequent
repetitions, many of the players played below the minimum effort level of the previous
repetition, which the authors referred to as overshooting, or, more descriptively, undercutting.
By the tenth period 72% of the subjects played the minimum effort level, although in all 70
repetitions Nash equilibrium never occurred. Of course, VHBB provides much more detail, but
21
this is the crux of the matter for the purposes of this paper. As is typical in economic
experiments, the payoffs are in cash, but this does not imply that the players have, or behave as if
they have, cardinal utility functions and the requisite knowledge of each other’s behavior.
This undercutting is the mechanism behind the downward cascade, the strong, and also
extremely robust, as it turns out, downward dynamic. To my knowledge, VHBB is the first
explicit recognition of this downward dynamic, an entirely “new” dynamic. While a great deal
of interesting work has been done on related specifications of coordination games, it is this
critical and perplexing downward dynamic that is examined here.
The approach taken here in qualitatively addressing the perplexing complicated behavior
and dynamics of VHBB’s results is not to deny the “draw” of Schelling’s focal Nash equilibrium,
the “draw” of Pareto dominance and Pareto optimality. But neither is it to deny the impact of
Morehous’s fear motive that characterizes the Stag Hunt game. Rather it is to recognize these
forces, but to observe that the player’s reaction to these different forces can well be idiosyncratic,
be expected to be idiosyncratic, and particularly so when the “draw” of rationality itself does not
bear on such idiosyncratic reactions, that is, when the game theoretic outcome is indeterminate.
In this scenario, it is the very dispersion of individual choices that is an ingredient in the
predictability of the joint behavior, the strong and extremely robust downward cascade.
In particular, with respect to the undercutting mechanism itself, it does not seem
unreasonable that some players would in repeated play find the minimum play of the previous
game focal, rather than, or more than, the Pareto dominant equilibrium, but would also be
“pulled down on” by Morehous’s fear motive. Then, as the play continued, the “draw” of the
focal would fall for many with their observation of the “downward cascades.” But the “draw” of
the focal Pareto dominant equilibrium still remains, with varying degrees of intensity, and
equilibrium does not obtain. Indeed, when the minimum effort is as low as possible, zero, the
relative benefit of successful coordination is highest. Further, it seems perfectly believable that
in a real world crisis, like the recent Financial Crisis and Great Recession, Morehous’s “fear”
motive might overpower the draw from focal equilibria, as it mostly does in the particularly
compact and clear formulation used in VHBB’s tacit coordination game experiments.
There is a further ingredient in the “downward cascades,” namely the above observation
that the “min rule” provides a “severe test of payoff dominance”” (VHBB, p. 236). In the case of
the “min rule” a downward deviation by a single individual at the minimum causes the same loss
to every other individual, no matter how large the team. But the larger the team, the more likely
a “low ball” player, a player particularly sensitive to Morehous’s “fear” motive; an aspect of
team size the other players likely recognize themselves. Team growth increases the “pull” of
Morehous’s fear motive if you will. The complementarity remains very tight in this “weakest
link” model.
If, on the other hand, as the team grows the effect of a single deviation lessens, this would
have an offsetting effect. The nature of the complementarity is, then, an important consideration,
as experiments verify, starting with Van Huyck, Battalio and Beil’s multiple player multiple
strategy “average opinion” experiments (1991). As an example, suppose the team product is
banking services. If the matter at hand involves possible bottlenecks in critical components in
delivering those services then the “min rule” would be appropriate. But if the matter is the team
aspect of, that is the positive externalities between the banks within, the banking system as a
22
whole, then the complementarity likely “loosens” as team, system, size grows. In general the
nature of the complementarity does significantly influence behavior, as Goeree and Holt’s (2005)
repeated experiments neatly verify, by varying the costs of own “effort” in the “min rule,” with
two strategies per player in pairwise matchings of players, that is, 2 player 2 strategy games. In
these repeated experiments, with low effort cost, the average of effort rises over time, and, with
high effort cost, the average of effort falls over time, but the dispersion of individual choices
remains.
There is a caveat concerning this effect of a “loosening” of the complementarity as team
size grows, however. In this scenario, lacking a generally applicable theory of the focal, one can
imagine that, if it is discernible, even a single individual’s decision could be focal for some. A
fear of “copy cats” itself could be self-fulfilling, to a degree at least. One might reasonably
conclude qualitatively, then, that the risk of coordination-failure is higher the larger the number
of players involved and the tighter the group’s complementarity. But the very possibility of
coordination-failure cannot be ruled out in any case. Moreover, when economies advance the
tightness of complementarities may grow, financial and real, within and between countries. If so,
the frequency and scale of crises may be expected to grow as well, all else equal.
23
ACKNOWLEDGEMENTS
I am very much indebted to Marc Dudey for bringing Rapoport and Guyer’s work to my
attention, to Carlos Zarazaga for valuable comments, and to Marcia Brennan, Richard Grandy,
Simon Grant, John Harsanyi, Herve Moulin, Martin Shubik, and Robert Solow for very
insightful observations, prior to this paper, concerning the application of game theory. Errors
and oversights are my responsibility alone.
24
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26