SELF-ASSEMBLING GAMES

JEFFREY A. BARRETT AND BRIAN SKYRMS

Abstract. We consider how cue-reading, sensory-manipulation, and signaling

games may initially evolve from ritualized decisions and how more complex

games may evolve from simpler games by polymerization, template transfer,
and modular composition. Modular composition is a process that combines

simpler games into more complex games. Template transfer, a process by which

a game is appropriated to a context other than the one in which it initially
evolved, is one mechanism for modular composition. And polymerization is

a particularly salient example of modular composition where simpler games

evolve to form more complex chains. We also consider how the evolution of
new capacities by modular composition may be more efficient than evolving

those capacities from basic decisions.

1. Introduction

Evolutionary game theorists analyze equilibria and dynamics of games that are

simple enough to be amenable to analysis. The question of the origin of the games

studied naturally arises. Social interactions seldom come as neatly structured as

the simple games used to model them. So where do the games come from?

On one reading, the question may seem superficial. If this is a problem for the

theorist, it seems to be in principle no more of a problem for the game theorist than

for any other modeler. A model of an airplane wing in flight, or of a bridge with

traffic and wind, is of necessity a simplification of the complex real system being

modeled. The theorist tries to capture the essence of the phenomena in a form

simple enough to be amenable to analysis. This can be done poorly or well—end

of discussion.

But there is another version of the question that is deeper. Games model re-

peated problems of interactive decision. We can ask: How do these games them-

selves form? How do the bits and pieces of decisions assemble into games? How

did individuals, in dealing with the world, come to interact in such a way that the

interactions can usefully be characterized as a game with specified players, strate-

gies, information, order of play and payoffs? And, more generally, how might such

games themselves evolve? These are theoretical questions worth investigating.

Our focus here will concern how decisions assemble into games, how games may

come to be appropriated to contexts that differ from those in which they are initially

played, and, in particular, how simple games assemble into more complex games.

Date: January 25, 2015.

1



2 JEFFREY A. BARRETT AND BRIAN SKYRMS

There are other aspects of the topic that call for consideration, but we take this to

be a suitable place to start.1

In what follows we consider a number of types of self-assembly. We concentrate

on signaling, although the general perspective applies to other kinds of interaction

as well. One might, for example, use the framework discussed here to consider a

public goods game evolving to work jointly with a bargaining game. We restrict our

attention to signaling games, however, for two reasons. To begin, signaling games

are well-studied and the considerations here help to unify recent work on signaling

systems in particular. But also, signaling games may evolve to implement simple

algorithms that might, by composition, evolve to implement more general forms of

computation. We will consider examples of this when we discuss the composition

of addition and ordering judgments and the composition of logical operations.

Throughout, some sort of adaptive dynamics will be fundamental to the analysis.

We will consider a small handful of different dynamical processes including simple

reinforcement learning, reinforcement with punishment, and reinforcement with

invention of new behaviors.2

There will also be a basic structure that characterizes each game. This structure

will specify the possible situations that may obtain, the stimuli to which the agents

are sensitive, the responses to which they are capable, and the rewards engendered

by the combination of actions in the situation at hand. The dynamics indicates

how agents’ dispositions to respond to stimuli evolve over time. The game structure

itself may evolve, as it does in the case of reinforcement with invention. And the

agent’s dispositions to respond may be appropriated to new tasks. These new tasks

may involve taking as input new aspects of nature or the actions of agents in other

games. In addition to evolving new games, the modular composition of games may,

under appropriate circumstances, provide more efficient paths toward the evolution

of complex dispositions.

2. The Assembly of Decisions into Games: Cue Reading, Sensory

Manipulation, and Signaling

Consider the origin of signals by the process of ritualization suggested by Tinber-

gen [1952], Lorenz [1966], and Huxley [1966]. Individuals of type S just happen to

produce a cue that individuals of type R learn to read. Then an adaptive dynamics

sets up a positive feedback that ritualizes the cue, so that it is used as a signal.

1Note that decisions, at least in the sense we have in mind here, are just agent actions or behaviors

that might be reinforced. As such, they presuppose no deliberation or even the capacity for rational
reflection.
2See (Erev, and Roth [1998]), (Barrett and Zollman [2009]), and (Skyrms [2010]) for discussions of
these and other closely-related evolutionary processes. Regarding the adaptive dynamics employed
in the models, other things being equal, we use the simplest that readily illustrates the particular

phenomena of interest.



SELF-ASSEMBLING GAMES 3

For instance, an animal may naturally bare its teeth before biting, with feedback

amplifying the behavior so that it becomes a ritualized threat signal. The process

may lead to complicated signals in simple animals. Scott, et al. [2010] investigate

complex vibratory signals of ownership in the caterpillar Drepana acurata. These

combine vibrations produced by anal scraping, using specialized organs, with those

produced by drumming or scraping their mandibles. Using comparative morphology

and comparative territorial behavior across many species, they conclude that anal

scraping is a ritualized form of the vibrations made by ordinary locomotion and

the vibrations produced by the mandibles are ritualized fighting behavior. In this

species they are combined to form a warning signal of ownership of territory that

will be defended.

Suppose that there is a situation where different potential ritualized signals may

be sent, depending on the state of the sender, and different behaviors may be

appropriate if the receiver reads those signals well. The situation might involve

fighting, mating, or anything else that might influence the actions of the receiver.

A particularly salient example is that of the predator-specific alarm calls that

are found in many species.3 In the process of becoming ritualized signals, a number

of decisions on the part of sender and receiver become bundled together as a game.

There is a state of the sender, which nature chooses according to some probability.

The sender performs an act that sends a signal, where the signal sent (or better, its

probability of being sent) depends on the sender’s state. The receiver observes the

signal and chooses some act with a probability that depends on the signal received.

The act has payoff consequences for both sender and receiver. The individuals

involved certainly need not think of this as a game; they may not think at all.

Signaling games may form in a variety of ways. There is evidence that the males

of some species may have evolved signal types that exploit preexisting female sen-

sory biases for the purpose of mating. One well-studied example is the Physalaemus

pustulosus species group of frogs. The females across the species in the group share

preferences for a particular set of call features. One of these, for example, is the

presence of low-frequency “chucks.” While males in one species may exploit one

of these preexisting preferences, males in a sister species may exploit another. It

is argued that the best explanation is that the males in each species have evolved

calls that exploit the shared preexisting preferences of the females and, as it hap-

pens, have ended up exploiting different preferences in the context of the different

species.4

3See, for example, (Cheney and Seyfarth [1992]) and (Manser et al. [2002]).
4See (Ryan and Rand [1993]). See also (Endler [1993]) and (Dawkins and Guilford [1996]) for
further discussion of the exploitation of preexisting sensory biases in the evolution of signaling.



4 JEFFREY A. BARRETT AND BRIAN SKYRMS

The formation of a signaling game by the exploitation of a preexisting sensory

bias might also be understood as ritualization. The female frogs are attracted by

particular sound, so the male frogs evolve to send mating signals that incorporate

that sound. Just as a receiver might evolve to exploit the fixed dispositions of a

sender to act in a particular way given the state of nature, here it is the sender who

evolves to signal in a way that exploits the receiver’s preexisting dispositions to act

in a particular way given, in this case, a particular type of sound. And a game may

be formed by the process of ritualization with positive feedback.

But to speak of “positive feedback” here is to trade specificity for vagueness, as a

number of mathematically precise realizations of evolution, imitation, and various

types of learning have been rigorously analyzed for a wide variety of simple signaling

games.5 In such games, the positive feedback mechanism is represented by a specific

evolutionary or learning dynamics that determines the payoffs for the senders and

receivers and how these payoffs influence their further actions. It is this dynamics

that may ultimately forge a ritualized signaling system. When it does, one has an

evolutionary model for the formation of that system.

Three simple two-agent games are suggested by such considerations. The first

is a cue-reading game, the second a sensory-manipulation game, and the third a

signaling game. In a cue-reading game, an agent evolves to take advantage of

nature’s fixed dispositions or the fixed dispositions of another agent who comes

to play the role of a sender. In a sensory-manipulation game, a sender evolves to

exploit the fixed dispositions of an agent who comes to play the role of a receiver.

And each of these games might be understood as simplified versions of a signaling

game where a sender coevolves to respond to nature and a receiver coevolves to

respond to the sender’s ritualized signals in a way that is advantageous to both.

In a two-agent cue-reading game, an actor performs some regular fixed action in

response to a particular corresponding state of nature. The exploiting agent then

observes this action and performs an action of his own. If the exploiter’s action is

successful, he reinforces his disposition to perform that type of action conditional

on the actor’s performing the corresponding action. And if it is unsuccessful, he

does not reinforce, and may even weaken, this conditional disposition. Here the

exploiter may quickly evolve to cue his actions to the behavior of the actor and

hence take advantage of the actor’s fixed dispositions to respond to nature.6

5See, for examples, (Hofbauer and Huttegger [2008]), (Argiento et al. [2009]), (Barrett and Zollman
[2009]), (Skyrms [2010]), (Hu et al. [2011]), and (Barrett [2013a]).
6Where the actor and exploiter have few available types of action, even simple Herrnstein rein-
forcement learning (Herrnstein [1970]) very quickly leads to successful cue-reading on simulation.

Note that the evolutionary task here is typically much easier than the task of coordinating signals
and acts in a signaling game inasmuch as the actions of the actor are already fixed in a cue-reading
game. The evolution of perception may be modeled by such a game. Indeed, a one-agent cue-

reading game, where the agent takes her cues from nature directly, provides a plausible account



SELF-ASSEMBLING GAMES 5

In a sensory-manipulation game, the sender evolves to takes advantage of the

fixed dispositions of the receiver in a similar manner. The sender observes the state

of nature and sends a signal. The receiver acts in a fixed way on the signal with

no choice. If the action is successful, the sender reinforces her dispositions to send

that type of signal on the present state of nature. And if is unsuccessful, she does

not reinforce, and may even weaken, this conditional disposition.

In a signaling game, the sender observes the state of nature, then sends a signal.

The receiver observes the signal, then performs an action. They may begin by

randomly signaling and acting. If the receiver’s action is successful, then the sender

reinforces the signal sent conditional on the state of nature that obtained and the

receiver reinforces his action conditional on the signal he observed. And if the action

is unsuccessful, the agents do not reinforce, and may even weaken, these conditional

dispositions. In some cases one can prove that such signaling games converge to

signaling systems, and in many cases, the agents evolve successful signaling on

simulation.7 Here the signals become ritualized through the reinforcement of the

sender’s dispositions to respond to nature with signals and the receiver’s dispositions

to respond to signals with actions.

As illustrated in Figure 1, in a Lewis signaling game the first player, the sender,

knows something that the second player, the receiver, does not know. This is

represented by nature choosing a state t1 or t2. The sender can only send an

arbitrary signal s1 or s2. The receiver observes the signal and takes an action.

The interaction is successful, and the payoff to each player is 1, if the receiver

chooses a1 when nature picked t1 or if the receiver chooses a2 when nature picked

t2. Otherwise, the payoff is 0.

A cue-reading game is a Lewis signaling game that has been pruned as indicated

by the shaded region of Figure 2. Here the sender has no choice and displays cue s1

in state t1 and no cue, which we will indicate by s2, in state t2. The receiver

observes the cue and takes an action.

A sensory-manipulation game is a Lewis signaling game that has been pruned

as indicated by the shaded region of Figure 3. Here the receiver has no choice and

does act a1 on signal s1 and act a2 on signal s2.

Just as the cue-reading and sensory-manipulation games can be understood as

simplified versions of a Lewis signaling game, they might also be understood as

for how the actor may have evolved her fixed dispositions in response to the various states of

nature in the first place. See (Isaac [2011], section 5.4.1) for a discussion of perception and the

information content of perception that fits well with a one-agent cue-reading model.
7See (Skyrms [2010]) for an extended discussion of such games and their properties.



6 JEFFREY A. BARRETT AND BRIAN SKYRMS

t1 t2

s1 s2

01

Nature

Sender

Receiver

Payoff1 1 10 0 0

s1 s2

a1 a2 a1 a2 a1 a2 a1 a2

Figure 1. A signaling game

t1 t2

s1 s2

01

Nature

Sender

Receiver

Payoff1 1 10 0 0

s1 s2

a1 a2 a1 a2 a1 a2 a1 a2

Figure 2. A cue-reading game

evolutionary steps in the direction of a signaling game as the agents’ decisions are

ritualized.8

Ritualization, then, may act to form cue-reading, sensory-manipulation, or sig-

naling games from decisions. Such games may, then, be extended as other agents

exploit established behaviors or form new signaling connections by ritualization. In

this sense, one might think of the ritualization of decisions as the glue that binds

agents to form simple games from their basic decisions, then increasingly complex

games from simple games.

8Cue-reading, sensory-manipulation, and signaling games may differ from each other by degree. A

cue-reading game where the sender’s initial dispositions are relatively fixed might look increasingly
like a signaling game if both the sender and receiver are rewarded for the receiver’s actions, further

reinforcing the sender’s initial dispositional bias.



SELF-ASSEMBLING GAMES 7

t1 t2

s1 s2

01

Nature

Sender

Receiver

Payoff1 1 10 0 0

s1 s2

a1 a2 a1 a2 a1 a2 a1 a2

Figure 3. A sensory-manipulation game

3. Polymerization

The optimal act on receiving a signal may include sending another signal to one

or more other agents. Suppose that some individuals learn to do this. Then, in

appropriate circumstances, they may spontaneously self-assemble to form chains of

senders and receivers to pass information along. For example consider alarm calls

that indicate that a dangerous predator is on the prowl nearby. Individuals who

have not seen the predator may repeat the alarm, and pass along the information.

Consider a simple homogeneous single-species signaling chain. The first member

of the chain spots the predator and gives the appropriate alarm call. This individual

is in a different game from the other participants—observing the predator and giving

the call. The others are all in the same game, observing a call given and giving a

call in turn. They have adopted the simple strategy of repeating the call. They

have self-assembled into a chain or a more complicated network, whose topology is

determined by the contingencies of space and time.

Signaling chains may even cross species lines. The Vervet monkeys studied by

Cheney and Seyfarth [1992] understand alarm calls of the Superb Starling. Hornbills

understand the alarm calls of Diana monkeys (Rainey et al. [2004]). Some bird

species have learned to understand each others alarm calls (Magrath et al. [2009]).

Here at least one species has a more complicated strategy in a more complicated

signaling game that takes as inputs both the alarm calls of other species and those

of its own, and outputs alarm calls of its own species. Then self-assembly proceeds

as before. Simple signaling modules are strung together to form signaling chains

and signaling networks.

The structure of the signaling network in these examples is evanescent, depending

on who is where when. But there are other examples where the structure of the



8 JEFFREY A. BARRETT AND BRIAN SKYRMS

network is endogenous. Given the costs and benefits, individuals who have learned

to react to a signal to their advantage and to “pass it along” may, for example,

self-assemble to form rings or stars.9

Ritualization of the sort that forms cue-reading, sensory-manipulation, and sig-

naling games in the first place may also do the work of connecting established

signaling modules. Suppose that a sender S and a receiver R may have established

a signaling system by ritualization. A new agent E1 who is sensitive to the signaling

actions of S may, by reinforcement learning on his own success and failure, learn

to read cues and benefit from S’s signals just as S and R do. Then another agent

E2 who attends to the actions of E1 may learn to read cues to his profit. And so

on. As new agents are added, the composite game grows in complexity and allows

for increasingly subtle connections between agents.10

The formation of signaling chains has been simulated.11 If one simply places

players in a long chain, and requires them to learn to signal de novo, it may take a

very long time.12 In some cases, however, the successful evolution of chains may be

certain to happen in the limit for chains of arbitrary length.13 And if individuals

have already learned their strategies in simpler contexts, then self-assemble, the

evolution of a chain can be quite rapid (Skyrms [2009]).

4. Template Transfer

Template transfer explains how a game that evolved in one context might come

to be used successfully in a new context.

When in equilibrium, the agents playing a signaling game have stable disposi-

tions. These stable dispositions might be understood as implementing a rule that

takes whatever stimuli to which the senders are sensitive as inputs, and which out-

puts the actions of the receiver. Template transfer occurs in a signaling game when

such an evolved rule is appropriated to a context different from that in which it

initially evolved. It involves the evolution of an analogy between the stimuli of the

old evolutionary game and a new set of stimuli that characterizes a new game. In

many cases the appropriation of an old rule to a new context may be significantly

more efficient than evolving a new rule from scratch.

9See (Bala and Goyal [2000]) for the description of a ring-forming game and (Huttegger and
Skyrms [2013]) for a proof that such networks form under the trial-and-error dynamics of probe

and adjust.
10Note that what passes as an agent in such games is very simple. It is nothing more than a

system that might condition its actions on the actions of other systems. As will be particularly
relevant in the next section, an agent may be profitably thought of as a functional unit, among
other functional units, in a single organism.
11See (Skyrms [2009]) for a discussion of how signaling chains may evolve.
12Even a chain of just three players where the new player is placed between the other two, takes

a very long time to evolve (Skyrms [2009]).
13For a discussion of this see Jonathan Kariv’s [2014] Ph.D. thesis on binary signaling chains.



SELF-ASSEMBLING GAMES 9

An example of template transfer is illustrated in the transitive-inference be-

havior of Pinyon Jays Gymnorhinus cyanocephalus and Scrub Jays Aphelocoma

californica.14

In an experiment reported by Alan B. Bond, Alan C. Kamil, and Russell P. Balda

[2003], seven stimulus colors were arranged in a random linear order that was fixed

for each bird. The birds were then presented with two keys, each illuminated with

a different color. If a bird pecked the key illuminated with the higher-ranked color,

then it was rewarded.

The birds were initially presented with only adjacent color pairs: red and green,

green and blue, . . . , or cyan and orange, and the position of the higher-ranked

stimulus randomized between left and right keys on each trial. New color pairs

were gradually added as the birds exhibited success in correctly selecting higher-

ranked colors. Each of the birds was eventually required to track all six adjacent

color pairs. The Pinyon Jays learned to choose the higher-ranked color better than

0.85 of the time. The Scrub Jays learned more slowly, but eventually reached a

similar level of accuracy.

The birds were then presented with nonadjacent color pairs to determine whether

the birds would order the nonadjacent color pairs based on what they had learned

from their experience with just the adjacent color pairs. Both species immediately

exhibited a high level of accuracy on the trials involving the nonadjacent colors,

where the experimenters understood accuracy as judgments that matched the color

order that the experiments had initially assigned to the colors. The Pinyon Jays

chose the higher-ranked color in the full ordering with an accuracy of 0.86 for

nonadjacent pairs, and the Scrub Jays with an accuracy of 0.77.

Interestingly, the two species exhibited different types of error and different la-

tencies depending on the particular elements in the full color ordering that were

presented. This suggests that the two species employ different mechanisms in mak-

ing their ordering judgments for nonadjacent pairs. Nevertheless, both species did

well judging nonadjacent color pairs after being trained on just adjacent color pairs.

The experimenters concluded that the birds were making transitive inferences

based on their experience in the first part of the experiment. But the birds were

doing more than that. In particular, since an ordering on adjacent color pairs does

not determine an ordering on non-adjacent pairs, the birds were imposing a linear

ordering on their learned ordering of adjacent colors. It is only by appropriating

a preexisting linear template that the birds could get from their experience with

adjacent color pairs to judgments that immediately agreed with the experimenters

14This section reports selected results from (Barrett [2013b], [2014a], [2014b]). See those papers
for further details regarding the model described in this section. See (Barrett [2013b]) as well for

an alternative model.



10 JEFFREY A. BARRETT AND BRIAN SKYRMS

predetermined full linear order. Indeed, that the experimenters themselves took

the birds’ judgments on nonadjacent color pairs to be correct and simply inferen-

tial suggests that the experimenters were also appropriating a preexisting linear

template to their understanding of the birds’ experience.

The evolution of an ordering rule and the subsequent appropriation of that rule

to order a new type of stimuli can be modeled by a simple signaling game and its

appropriation to a new context.

Consider a signaling game with two senders, A and B, and a receiver R. The

senders and receiver may be understood as functional elements of a single bird.

Nature chooses at random and without bias two stimuli from a set of seven, and

one is presented to each sender, a to A and b to B. The stimuli, which might

be represented by the natural numbers 1 through 7, are ordered, with the order

represented in the payoffs of the game. The two senders react to the stimuli by

sending signals to the receiver, who performs one of three types of act: (0) a > b,

(1) a < b, or (2) a = b. The receiver’s act will count as successful if and only if

it correctly represents the predetermined order of the stimuli to the two agents A

and B. The agents evolve an ordering template here if they coevolve signals that

represent the possible stimuli and a dispositional rule for linearly ordering them

that typically produces successful actions by the receiver.15

Suppose that the agents learn by reinforcement with invention.16 On this dy-

namics, one might imagine that each sender has an urn corresponding to each

possible stimulus and that each urn begins with just a single black ball. When

presented with a particular stimulus, each sender draws a ball at random from the

corresponding urn. If the ball is black, a new signal type is invented and sent to the

receiver; otherwise, a signal of the type of the drawn ball is sent. The receiver also

has an urn corresponding to each ordered pair of signals he might receive. And if

he gets a new signal type, he introduces corresponding urns to represent the new

possible pairs. Each of the receiver’s urns begins with a single ball of his three

act types: a > b,a < b, and a = b. If successful, the ball drawn from each urn is

returned and a new ball of the signal or act type used in that play of the game is

added to the urn; otherwise, the ball drawn from each urn is just returned. Finally,

newly invented signal types are only kept and reinforced if they lead to a successful

act the first time they are used. Thus the game itself evolves.

15Compare to Ariel Rubenstein’s discussion of linear order in the second chapter of (Rubenstein
[2000]).
16See (Skyrms [2010]), (Argiento [2009]), and (Alexander et al. [2012]) for descriptions of this

dynamics and its properties. We consider this adaptive dynamics here as it shows how the agents

might both invent a representation for the stimuli and develop successful ordering dispositions
with modest evolutionary resources. This invented representation is then appropriated to a new

task.



SELF-ASSEMBLING GAMES 11

On simulation, the senders begin by quickly inventing an assortment of new

signals that they initially send more or less randomly. Consequently, the receiver

initially acts pretty much at random. After 107 plays, however, the cumulative

success rate is typically (0.99) better than 0.75; and, in general, the more plays,

the better the cumulative success rate.

The composite system might be thought of as having evolved to implement

a dispositional rule that takes naturally ordered stimuli as input, represents the

stimuli as signals, then outputs an act that reliably indicates the natural order of

stimuli. Once evolved, this dispositional rule might be appropriated as a template

to represent and judge the natural order of a new type of stimuli.

Such a template might be fit to a new context by coordinating the new stimuli

with the old inputs to the dispositional rule. The association of the new stimuli

with the old inputs to the dispositional rule might be thought of as implementing

an analogy between the new and old stimuli. When such an analogy evolves, the

old dispositional rule evolves to treat the new stimuli similarly to how it treated

the old stimuli that were involved in forging the old dispositional rule. This sort of

template transfer might be evolutionarily favored when the process that coordinates

the new stimuli to the old inputs is more efficient than evolving a new rule for the

new context.

We investigate this possibility by considering a simple model. Consider a new

ordered set of stimuli represented by the first seven letters of the greek alphabet α

to η. Suppose that the agents in the last game have already evolved a dispositional

rule for representing and ordering the old stimuli 1 to 7, but that they must learn to

represent and linearly order the new greek stimuli to be successful. The old evolved

dispositional rule might be used as a template for ordering the new stimuli if the

agents can evolve to associate the new stimuli to the old inputs in a one-to-one

way.17

Suppose that nature chooses at random and without bias two stimuli from the

set α to η, and one is presented to each sender A and B. Each sender has an

urn corresponding to each of the new stimuli types (Figure 4). Each of these urns

initially contains one ball corresponding to each of the old sender urns 1 to 7. When

presented with a new stimulus, each sender draws a ball at random from one of

her new greek urns, then draws a ball from the old arabic urn indicated on the

ball from the first draw, then sends the signal indicated on the second draw to the

receiver. Since the agents have already evolved a dispositions that order the old

stimuli (and hence the old arabic urns), the agents just need to learn to associate

17If there were fewer new stimuli than old stimuli, the agents might be successful even if their
association of the new stimuli to the old inputs was not one-to-one, but still respected the natural

order of the new stimuli.



12 JEFFREY A. BARRETT AND BRIAN SKYRMS

the new greek urns with the old arabic urns in a way that respects the new linear

order to be successful on the new stimuli.

Nature Translation urns Old urns

Stimulus a

Stimulus b

α β γ δ

ηζ

α β γ δ

ηζ

1 2 3 4

5 6 7

1 2 3 4

5 6 7

a > b

a < b

a = b

Act

Figure 4. A simple example of template transfer

We will suppose that the agents learn on this new game by simple reinforcement

with punishment, where success is determined by whether the results of the re-

ceiver’s judgments match the natural order of the new stimuli.18 More specifically,

if, on a particular play of the game, the agents choose balls from greek urns that

indicate arabic urns that produce signals that lead the receiver to correctly order

the new stimuli, then each sender returns her ball to the greek urn from which it

was drawn and adds a copy of the same ball type; otherwise, she discards the ball

she drew unless it was the last ball of its type in the urn, in which case, she simply

returns it to the urn.19

On simulation, the agents typically (0.995 on 1000 runs) evolve to successfully

match the new greek stimuli to the corresponding old arabic ordering system with

an accuracy better than 0.80 with 105 plays per run. Here the receiver’s dispositions

are already well-tuned to making successful linear ordering judgments on the old

stimuli. The senders, then, just need to evolve a successful analogy between the

old and new stimuli.

With respect to modeling the behavior of the jays, when the old dispositional rule

evolves in the context of ordering arabic inputs and the new association urns are

18Our strategy regarding the adaptive dynamics is to use the simplest sort for the explanation at

hand. Using reinforcement with invention to evolve the ordering dispositions in the basic game
explains how the birds might invent an internal representation and the associated dispositions.
Reinforcement with punishment explains how this invented system might be transferred to a new

context on a simple dynamics.
19Note that the contents of the old arabic-ordering urns do not change on plays of this game, and,

hence, the template is assumed to be fixed. One could allow the template to itself evolve as it is
transferred. This might better tune the template to the new context, but that would also involve

a different game than the one discussed here.



SELF-ASSEMBLING GAMES 13

trained on just adjacent greek inputs, the new composite system evolves to judge

the full order of greek stimuli with an accuracy better than 0.80. Here the composite

system sometimes evolves to match greek to arabic stimuli in a one-to-one manner

that respects the full arabic ordering, but more typically the arabic ordering rule is

used to group the greek stimuli into two or three linearly ordered segments. Within

each segment, the composite system does very well in ordering both adjacent and

nonadjacent greek stimuli. Between segments, the judgments are less reliable, but

in aggregate, the composite system does about as well as the judgments of the jays

when they apply a prior linear-order template to their experience of only adjacent

color pairs.20

5. Modular Composition

Modular composition occurs when a game comes to accept the play of another

game as input, thus forming a composite game. Here the stimuli to which the

agents in one game are sensitive are the actions of agents playing other games.

Polymerization is a special case of modular composition. More generally, modu-

lar composition of simple games may lead to heterogeneous networks of arbitrary

topology.

There are two aspects to modular composition. The first concerns how the inputs

to the component modules are determined. The second concerns how modules come

to interpret the actions of other modules to allow for successful coordinated action.

Concerning the first, the natural saliences for an agent are determined by the

perceptual apparatus of the agent. For the purposes at hand, we will simply stipu-

late what each agent responds to, then consider how they might evolve to use these

inputs and other evolved dispositions for successful action. Of course, an agent’s

natural saliencies might themselves evolve on feedback from her success and failure

in action.

Concerning the second, games may come to interpret the actions resulting from

the play of other games by the same reinforcement mechanism that allows for

template transfer. In particular, the glue that binds a simple game to other games,

is an evolved analogy between how the game treated its old inputs and how it treats

new inputs resulting from the play of the other games. Here is a rich experimental

example, accompanied by a simple computational model.

In a recent series of experiments, Livingstone et al. [2014] trained Rhesus macaque

monkeys to associate the number of dots on either side of a touch screen with the

number of food drops of reward the monkey would get if it selected that side of the

20See section 6 of (Barrett [2014b]) for further details regarding how an old ordering template
evolves to order a new type of stimuli on the incomplete evidence of only adjacent pairs of the

new stimuli.



14 JEFFREY A. BARRETT AND BRIAN SKYRMS

screen. They then trained the monkeys to associate symbols with the number of

food drops, to add the symbolically represented reward magnitudes, and to trans-

fer their learned arithmetic competence to a new set of symbols. We will briefly

discuss the experiments, then model the behavior of the monkeys by the modular

composition of two games. One game will involve ordering judgments, the other

addition.

The first experiment was a dots comparison task. Here the monkeys were pre-

sented with two circles on either side of the touch screen each containing randomly

placed dots of various sizes and colors. When a monkey touched a side of the screen,

it was rewarded with the number of liquid drops corresponding to the number of

dots on the side it chose. The device that fed the monkeys distinctly beeped once

as each drop was dispensed. The monkeys learned to choose the option with the

greater number of dots with an accuracy between 80% and 90%.

They were then trained on the first symbol comparison task. Here the experi-

menters associated a particular symbol with each cardinality from 0 through 25.

The symbols were neither cognates for the numbers nor did they exhibit any spe-

cial pattern. The monkeys were then presented with a symbol on each side of the

touch screen and were rewarded with a number of drops, each accompanied by a

beep, corresponding to the predetermined value of the symbol they selected. The

monkeys learned to choose the option with the greater number of dots with a high

degree of accuracy, particularly when there was a significant difference between the

value of the two symbols.

In the dots addition task, the monkeys were presented with one set of dots on

one side of the screen and two sets of dots on the other side of the screen, where

each set was specified by a circle around the dots. They were then rewarded in

drops, accompanied by beeps, for the total number of dots on the side of the screen

that they selected. Here the monkeys learned to reliably pick the larger of the

two options when the sum of cardinalities of the sets on one side of the screen was

significantly different from the cardinality of the set on the other. When the two

options were close in value, their behavior approached chance.

For the first symbol addition task, the monkeys were presented with two of the

symbols they had learned in the first symbol comparison task in an oval on one side

of the screen and a single symbol on the other side of the screen. The monkeys were

then rewarded in drops, accompanied by beeps, equal to the sum of the symbols in

the oval or the value of the single symbol respectively. Significantly, the monkeys

initially acted as if they were comparing the larger of the two addends against the

symbol on the other side of the screen. But they eventually learned to combine the

magnitudes represented by the two addends and reliably pick the larger of the two



SELF-ASSEMBLING GAMES 15

options when their sum was significantly different from the cardinality represented

on the other side of the screen.

Once the monkeys were successful at this task, a new set of symbols was intro-

duced in the second symbol comparison task. They successfully learned this new set

of symbols by comparison just as they had learned the first set of symbols in the

first symbol comparison task.

Finally, in order to determine whether the monkeys could transfer their arith-

metic competence with the old symbol set to the new symbol set, they were pre-

sented with a second symbol addition task where they were required to add the new

symbols just as in the first symbol addition task. The experimenters argued that if

the monkeys could learn to do so more efficiently than in the first symbol addition

task, this would indicate that they were able to transfer their arithmetic compe-

tence to a new system of representation, which might then be taken as evidence

that they were using the symbols as representations to carry out computations and

not just memorizing particular symbol combinations.

In the first symbol addition task, the monkeys reached a stable success rate after

about 50 days, but when presented with the second symbol addition task, the mon-

keys immediately chose the larger side more often than they did at a comparable

time during the first symbol addition task, and their performance reached a stable

success rate in just 10 days. The experimenters interpreted this as evidence that

the monkeys were able to transfer their previously learned competence at symbolic

arithmetic to a new context, and, hence, were carrying out computations on the

cardinalities represented by the symbols and not just memorizing symbol combina-

tions.

While it is unclear precisely what the monkeys were doing cognitively in each

task, one might model the evolution of such arithmetic competences using signaling

games, template transfer, and modular composition.

The dot comparison and the first and second symbol comparison tasks involve

the monkeys learning orderings of collections of dots and symbols just as the jays

learned orderings of colors. This can be modeled by a two-sender signaling game

similar to the one discussed in connection with the jays. Here each of the two

senders in the model, again, thought of as different functional elements of a single

animal agent, would have access to one side of the screen.

Template transfer illustrates how an ordering rule that evolved in one context

might come to be used successfully in another context, as in the second symbol

addition task where the monkeys learn to use the second symbol set in a way anal-

ogous to the first symbol set. Indeed, by means of template transfer, the monkeys

might learn the second symbol addition task more efficiently than they learned the

first symbol addition task without learning to carry out arithmetic computations.



16 JEFFREY A. BARRETT AND BRIAN SKYRMS

More specifically, they might learn successful combinations of symbols from the

first symbol set, then evolve an analogy between the old symbol set and the new

symbol set by template transfer.

If so, this poses a challenge to the experimenters’ conclusion that the second

symbol addition task involves computation. On the model just described, the com-

petence of the monkeys in the second symbol addition task is achieved by memo-

rization and template transfer, not computation. Further, template transfer here

also explains why the monkeys learned the second symbol addition task faster than

the first. On the other hand, transferring a learned rule of association from the

context in which it was learned to a new context by analogy might count as evolv-

ing a very basic type of concept. Here these concepts might represent cardinalities

and how they might be ordered.21

There are several ways that the monkeys might evolve to order dot-sums and

symbol-sums in the first place. They might evolve the ability to add the cardinalities

and order them all at once. Or they might learn how to add and how to order

separately, then learn how to compose these skills to carry out the addition-ordering

tasks set for the monkeys here.

In this second case, the dot addition and the first symbol addition tasks might be

modeled as the modular composition of two, more basic, evolutionary games. More

specifically, one might consider the composition of a two-sender ordering game very

much like the one discussed in the last section and a two-sender addition game that

evolves to compute the sums of the cardinalities presented to each sender.

We will consider the two-sender addition game first, then consider how that game

might evolve to communicate successfully with the two-sender ordering game.

In the two-sender addition game each sender is presented with a random set of

zero to five dots (or, equivalently, one of six symbols) where each cardinality has

equal probability. The senders then each send a signal to the receiver who performs

an action that depends on the signals. If that action corresponds to the sum of

cardinalities of the dots presented to the senders, then that particular play was

successful and the three agents, who can together be thought of as functional parts

of single composite agent, reinforce the conditional actions just taken.

We will suppose, as we did with the evolution of the ordering rule in the last

section, that the agents learn here by reinforcement with invention.22 Each sender

has six urns, one for each cardinality zero to five. Each urn initially contains just

21See (Barrett [2014b]) for a discussion of how the evolved appropriation of a rule might be

understood as the evolution of increasingly general concepts in the context of the ordering game
discussed above.
22Just as in the case of the order rule in the last section, using reinforcement with invention as
the adaptive dynamics here explains how the an appropriate representation of states and actions

might be invented.



SELF-ASSEMBLING GAMES 17

a single black ball. While there is always precisely one black ball in each of the

sender’s urns, other balls, labeled with terms corresponding to potential signals,

are added to the urns on repeated plays of the game. When presented with a set

of dots, each sender draws a ball at random from the corresponding urn. If the

ball is black, the sender invents a new term at random and sends that term to the

receiver; alternatively, if the ball is labeled with a term, the sender simply sends

that term to the receiver. On each play of the game, then, the receiver gets one

signal from each sender.

The receiver has an urn labeled by each pair of terms that he has received from

the senders so far, and he constructs further new urns as needed as the senders

invent and send new terms. Each of the receiver’s urns begins with eleven balls, one

ball corresponding to each possible sum action from zero to ten. When the receiver

gets the pair of signals from the sender, he draws a ball at random from the urn

corresponding to that particular pair of signals, then carries out the corresponding

act.

If the act matches the sum of the cardinalities of dots and neither sender drew

a black ball, then each agent returns the ball he drew to its urn and adds a copy

of that ball. If a sender drew a black ball and the receiver’s action was successful,

then the sender adds one ball of the new signal type he invented to each of his

urns, and the receiver adds urns corresponding to the new types of signals he may

receive. If the receiver’s action was not successful, the agents just return the balls

they drew to the urns. Note that once a new term is invented, the agents learn by

simple reinforcement without punishment.

On simulation, since the senders do not initially have much in their urns, they

draw the black ball often and, hence, begin by inventing new terms at a relatively

high rate. While they initially send the newly-invented terms at random, and the

receiver initially acts randomly when he receives a new pair of terms, as the agents

update their first-order dispositions by reinforcement, they typically evolve a set of

systematically interrelated dispositions where the receiver’s act corresponds to the

sum of the cardinalities presented to each of the senders.23 Here the senders’ terms

have evolved to represent cardinalities and the receiver’s dispositions have evolved

to add those cardinalities.

As with the birds, the agents in the game might be taken to represent various

functional parts of a single monkey. On this interpretation, the model illustrates

23In particular, on 1000 runs of 8.0 × 106 plays each, the cumulative success rate is greater than
0.95 in 0.67 of the runs, greater than 0.90 in 0.83 of the runs, and greater than 0.85 in 0.95 of the

runs.



18 JEFFREY A. BARRETT AND BRIAN SKYRMS

how a monkey might coevolve an internal representation of cardinalities and the

ability to reliably calculate sums on the basis of this representation.24

Given that the monkeys possess such addition dispositions, which, as we have

just seen, might themselves evolve in the context of a signaling game, one might

model the behavior of the monkeys in the dot addition and the first symbol addition

tasks as the modular composition of a two-sender ordering game and a two-sender

addition game. Here the addition game takes its two inputs from the two sets of

dots on one side of the screen, then the ordering game takes one input from the

output action of the addition game and the other input from the dots on the other

side of the screen. If these games can evolve to communicate successfully, the result

will be a composite game where the monkeys sum the cardinalities on one side of

the screen, then compare this sum against the cardinality on the other side of the

screen and select the larger result.

Suppose that the addition game has evolved to compute the sum of inputs a and

b from the left side of the screen and produce an act corresponding to the sum.

And suppose that the comparison game has evolved to choose the greater of two

cardinalities l and r and is sensitive to the act a + b of the addition game and

the cardinality of the righthand side of the screen r. The comparison game has

evolved to add cardinalities of sets of dots, so, at least initially, it does not know

what to do with the act a + b of the addition game. In order to work together,

an association must evolve between the two modules. This can occur as a form of

template transfer.

When a set of dots is presented to the R input, the sender in the comparison

game sends whatever signal it has evolved to send. But when an act from the

addition game is presented to the L input of the comparison game, it is matched

up, at least initially, with a random signal.

Just as in the bird ordering template-transfer model, we will consider how the

analogy might evolve by reinforcement with punishment. Consider eleven transla-

tion urns labeled 0 to 10 corresponding to each of the possible act types resulting

from the addition game. Each urn initially has one ball each of each signal type

used in the comparison game labeled 0 to 10. When an action is presented to the

L input of the comparison game, a ball is drawn from the corresponding urn and

the signal corresponding to that ball is sent to the receiver of the comparison game

(Figure 5). If the comparison game judges the relative cardinalities of a + b and

r correctly, where either a + b > r or a + b < r in each case, then the translation

ball drawn is replaced in the urn from which it was drawn and a copy is added;

24See (Barrett [2013a]) for a model where the agents evolve the ability to add increasingly large
cardinalities. The sense in which this counts as an arithmetic computation is relatively weak, but

insofar as the agents can learn to transfer this rule to other contexts, it might, again, be thought
to represent a potentially general rule and hence an arithmetic computation.



SELF-ASSEMBLING GAMES 19

otherwise, the ball drawn is discarded unless it is the last ball of its type in that

urn.

Screen Modules Act

R

L

r

a

b

translation urns

a + b > r

a + b < r

comparison
game

addition
game

Figure 5. Modular composition of addition and ordering

On simulation, the addition and comparison games are found to be successfully

coordinated by reinforcement learning with punishment. With 1000 runs of 105

plays each, the composite system evolved to correctly order a + b and r with an

accuracy typically about 0.94 and always better than 0.90. And when it gets the

order wrong, the mean difference between a + b and r was 1.52. This agrees well

with the monkeys’ degree of success, and with their being more likely to make a

mistake when the cardinalities being compared are close.25

Part of the work is done by how the two modules are connected. Here we simply

stipulated in the specification of the composite game the stimuli to which the inputs

to the addition and ordering rules were sensitive without saying how the modules

might have come to be sensitive to those particular types of stimuli. There are,

however, considerations that may serve to limit the options.

In the case of chains of alarm calls, the question of what each agent attends to

was unproblematic. Alarm calls have been learned to deserve attention in simple

contexts, and one might expect this salience to transfer to more complicated chains

25Indeed, one should expect the latter phenomenon when a skill evolves by reinforcement on

ordering judgments since the composite agent needs not get the order exactly right to enjoy a

high degree of success. One might, consequently, expect the monkeys to exhibit more precise
addition behavior in an experiment where the success of their action on each play depends on

getting the sum exactly right rather than just getting an order judgment right.



20 JEFFREY A. BARRETT AND BRIAN SKYRMS

or networks of games. Further, the same features that make for a good signal,

chiefly that it is easily noticed given the de facto perceptual capacities of the agents,

may also make it salient to the agents responsible for the input to other modules.

More generally, the actions resulting from evolved modules are actions that have

mattered to payoffs. Insofar as agents attend to payoffs, then, such actions might

be similarly salient in similar contexts.

In short, whether it is more efficient to evolve a new capacity from scratch or by

modular composition will depend to a large degree on precisely what is salient to

the agents and how.

6. Relative Efficiency of Template Transfer

The modules in the bird transitive inference and monkey addition examples

evolved on reinforcement with invention and template transfer evolved on rein-

forcement with punishment. In order to compare the relative efficiency of template

transfer as directly as possible, we will consider a very simple logical game that

evolves on reinforcement learning, then compare this to template transfer on pre-

cisely the same learning dynamics.

The nand game is a two-sender, one receiver signaling game where the agents

learn by simple reinforcement. Nature randomly and independently picks a state 0

or 1 for each sender A and B. Let SA and SB be these states. Each sender has one

urn for each of the two possible states, and each urn starts with one ball indicating

signal g and one indicating signal r. The receiver R has one urn for each pair

of signals he might receive from the two senders: [gA,gB], [gA,rB], [rA,gB], and

[rA,rB], and each of these urns starts with one ball labeled 0 and one ball labeled

1. A play of the game is successful if and only if R performs the act corresponding

to SA nand SB; that is, if R does act 0 when both SA and SB are 1 and does act

1 otherwise. If a play is successful, each agent returns the ball she drew to the urn

from which it was drawn and adds another of the same type; otherwise, each agent

simply returns the ball she drew.

The nand game typically coevolves a basic signaling language and the operator

nand. On 1000 runs with 106 plays/run, 0.71 of the runs exhibit a cumulative

success rate of better than 0.80, 0.62 of the runs better than 0.90, and 0.50 of the

runs better than 0.95. The game sometimes does not evolve nand. When it fails to

do so, every pair of states produce act 1, and the game is, consequently, successful

on about 0.75 of the plays.26

Once nand has evolved, it might be appropriated to a new context by template

transfer. For the purposes of direct comparison, we will also consider template

transfer in the context of simple reinforcement learning.

26This happens on about 0.23 of the runs with 107 plays/run.



SELF-ASSEMBLING GAMES 21

Suppose that nature presents each sender with a random state 0′ or 1′. Each

sender has two translation urns, one labeled 0′ and one labeled 1′, that each start

with one ball labeled 0 and one labeled 1. Each sender draws from the translation

urn corresponding to the current state presented to her, draws from the old urn

indicated on the ball drawn from the new translation urn, then sends the old signal

indicated by that draw to the receiver who, then, acts with whatever dispositions

he acquired when nand evolved in the old context. A play of this game is successful

if and only if the receiver preforms the act corresponding to SA
′ nand SB

′. If the

play is successful, each of the senders returns the ball she drew to the translation urn

from which it was drawn and adds another ball of that type; otherwise, each sender

simply returns the ball she drew. The receiver does not change his dispositions.

This template transfer game typically appropriates nand to the new context an

order of magnitude faster than the initial evolution of the logical operator on the

same dynamics. It does so by evolving a map from new inputs to old signals that

exploits the fact that the receiver’s dispositions have already evolved to calculate

nand on those signals. On 1000 runs with 105 plays/run, 0.78 of the runs exhibit a

cumulative success rate of better than 0.80, 0.61 of the runs better than 0.90, and

0.50 of the runs better than 0.95, which is about the same level of success that the

original nand game achieves on 106 plays/run.27

Template transfer also allows for a more subtle sort of efficiency. Once nand has

evolved, it may be appropriated to a new context by template transfer to play the

role of a different logical operator more efficiently than that operator might evolve

on its own. We will consider one example of this.

The operation or typically evolves on simple reinforcement learning just as in the

nand game. And, perhaps unsurprising given the similarity in their truth tables,

its evolution exhibits about the same efficiency on simulation.

Consider the template transfer of nand into a context where a play is successful

if and only if the receiver preforms the act corresponding to SA
′ or SB

′, and sup-

pose that the senders update their translation urns by simple reinforcement. On

simulation, the composite system evolves the logical operation or from nand an

order of magnitude faster than or evolves on the same dynamics. Template trans-

fer accomplishes this by mapping new inputs to old signals in such a way that the

receiver produces his 0 output in state (0, 0) instead of (1, 1). This works because

both logical operations produce 0 on exactly one pair of inputs.28

27For further comparison, on 105 plays/run in the original nand game, 0.60 of the runs exhibit

a cumulative success rate of better than 0.80, 0.48 of the runs better than 0.90, and 0.34 of the
runs better than 0.95. Like the nand game, the template transfer game also fails to evolve nand

for the new inputs on about 0.23 of the runs with 107 plays/run.
28This works the same way for the other two binary logical operators that produce one false. More

generally, if one had five binary logical operators, one for each possible number of false outputs,



22 JEFFREY A. BARRETT AND BRIAN SKYRMS

7. Discussion

The ritualization of decisions explains how cue-reading, sensory-manipulation,

and signaling games may initially evolve and and how new games may evolve from

old by polymerization, template transfer, and modular composition. These pro-

cesses are interrelated.

The ritualization of decisions explains how cue-reading, sensory-manipulation,

and signaling games may get started. An agent may condition his actions on some

aspect of nature, then learn from the consequences. An agent may condition his

actions on some aspect of nature and another condition her actions on the actions

of the first agent, then each learn from the consequences. An agent may learn to

manipulate the actions of another agent by taking advantage of the fixed dispo-

sitions of the second agent. Or agents may coevolve dispositions that coordinate

their actions to their mutual benefit. In each of these cases, the decisions that

result from the de facto dispositions of the agents form a game by ritualization.

Such games may eventually lead to stable, successful dispositions that implement

a rule. And such modular rules may, in turn, compose with each other by further

ritualization of decisions.

Polymerization is a simple case of modular composition. Chains of alarm calls

provide a compelling example of the sort of natural saliences that may serve to

connect modules that initially evolved independently. Given appropriate saliences,

ritualization of the sort that leads to cue reading, sensory manipulation, and signal-

ing explains how such modules might compose to allow for successful coordinated

action. The process of template transfer, in the examples described here, is a special

type of cue-reading that is particularly well suited to modular composition. And,

while the examples here involve a type of cue reading, template transfer might also

compose modules by the ritualization of decisions between agents akin to those that

form sensory-manipulation or signaling games.

Template transfer explains how a game or rule may come to function in a context

different from that in which it initially evolved. The appropriation of the module

to a new context involves the evolution of an analogy between the old inputs to the

game and the new inputs. The new inputs might come from nature or from the

actions of other agents. In the later case, a module may learn how to interpret the

actions of other modules as inputs to produce successful actions. Such functional

composition between modules may be more efficient than evolving the new capacity

from scratch.

one could get the other eleven by template transfer. And each would evolve an order of magnitude
faster this way than on its own.



SELF-ASSEMBLING GAMES 23

The evolution of strategies in a given game is a vibrant area of ongoing research.

But the question of the evolution of games themselves is important and deserves to

be explored. Here we have taken a few initial steps.

8. Acknowledgements

We would like to thank Louis Narens, Simon Huttegger, Kevin Zollman, Kai

Wehmeier, Alistair Isaac, Cailin O’Connor, and the two anonymous reviewers for

helpful comments. We would also like to thank Thomas Barrett for producing the

figures.



24 JEFFREY A. BARRETT AND BRIAN SKYRMS

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UC Irvine; Irvine, CA 92697, USA

E-mail address: j.barrett@uci.edu, bskyrms@uci.edu