1. INTRODUCTION
Neuroeconomics mainly focused on economic experiments using methods of brain imaging (for surveys see Glimcher and Rustichini Reference Glimcher and Rustichini2004; Camerer, Lowenstein and Prelec Reference Camerer, Loewenstein and Prelec2005; McCabe Reference McCabe2008). Since neural activity is not explicitly modelled in economic theories, such theories may be of limited use for generating hypotheses that guide neuroeconomic experiments in an insightful way. To fill the gap, neuroeconomic theories are required that are more explicit on the biological constraints that the nervous system imposes on behaviour. In developing such theories, the formal tools of game theory may be a useful language for modelling complex phenomena of interaction within and between brains as it was similarly useful in the development of modern economic theory. The aim of this note is to outline how existing tools of evolutionary game theory and learning in games may be reinterpreted to shed some light on the development of “brain” functions in a changing environment. No claim of originality is made: the main result has been developed elsewhere in the abstract context of evolution and learning in games by Germano (Reference Germano2007).
We consider a finite set of players who play repeatedly different strategic games selected randomly according to some exogenously given probability distribution on a finite set of games. The players are endowed with a “nervous system”. This is a suggestive interpretation of a simple network-like structure with “neurons” as nodes and “synapses” as binary relation on neurons. The structure constrains the player's perception of the environment and her behavioural response – similar to incomplete information in games. The “richer” the nervous system, the better it can detect the variability of the environment and the more variability of behaviour it can generate. We ask the following question: Can such a nervous system be designed by evolution, development and learning to “function well” in the player's interaction with other players and the environment? Intuitively, a “well functioning” brain should be adapted to its environment in the sense of generating appropriate behavioural responses that enable the survival of the population of “brain-carriers”. In this paper, we assume that “function well” means the brain's ability to play strategies that are not strictly dominated in the respective games and in the “average” game over the player's life-time. Reinterpreting a result by Germano (Reference Germano2007), we answer this question affirmatively. Yet, a simple example shows that if players can endogenously affect the change of the environment (like in non-trivial stochastic games), then this conclusion may not hold anymore.
At first glance, the evolutionary approach sketched in this note seems to be orthogonal to “mainstream” neuroeconomics today but we argue that it is relevant for the foundations of neuroeconomics. While economics studies optimal decision making, a typical neuroeconomic experiment will produce brain images of subjects when confronted with an economic decision task. These data are then interpreted with constructs that play a role in economic theories such as utility, expected utility, multiple selves etc. despite the fact that economic theory treats those as abstract constructs and optimizing behaviour “as if”. So the implicit assumption in neuroeconomics is that the brain is the very machine that could produce in principle optimal or constrained optimal behaviour. More generally, the assumption behind functional magnetic resonance imagining (fMRI) is that different subsets of the brain are activated to fulfil different functions or goals. Glimcher (Reference Glimcher2003: Chapters 6 to 8) traced this assumption back to Marr, who according to Glimcher (Reference Glimcher2003: 142) suggested that “(i)n order to understand the relationship between behavior and brain, one had to begin by understanding the goals or functions of a behavior. Then one could begin to ask how the brain accomplishes a specific goal.” Further he writes (p. 167)Footnote 1 that “(t)he goal of the nervous system is to maximize the inclusive fitness of the organism”. The question that we raise in this paper is whether or not evolution, development and learning can produce a nervous system that is capable of doing that. This answer seems to be not obvious to neuroscientists. According to Glimcher (Reference Glimcher2003: 166) a “major criticism that Marr's approach has faced is that it has been unclear whether evolution can be conceived of as a process that structures nervous systems to accomplish goals with enough efficiency to make the computational goal a useful starting point for neurobiological analysis.”Footnote 2 This note may be seen as a very preliminary attempt to provide an answer to this criticism of the foundations of neuroeconomics with some tools of evolutionary game theory.
We are not the first to sketch some neuroeconomic theory. Others realized that hypotheses on how the brain constrains economic behaviour should be ideally grounded on models that integrate microeconomic theory with a theory of the brain. Recent papers by Benhabib and Bisin (Reference Benhabib and Bisin2005), Bernheim and Rangel (Reference Bernheim and Rangel2004), Brocas and Carrillo (Reference Brocas and Carillo2008a, Reference Brocas and Carillo2008b), and Fudenberg and Levine (Reference Fudenberg and Levine2006) build models with “multiple selves” motivated by the modularity of the brain but do not really attempt to represent physiological elements of the brain.Footnote 3 Hence, they are of limited use for generating hypotheses on neural data observable with modern technology (while being capable of generating hypotheses on economic behaviour).Footnote 4 Our approach here is different in that (besides taking an evolutionary approach) we seek to complement standard game theoretic models with a (crude) model representing physiological elements of the brain such as neurons and synapses. The hope is that an enhanced version could generate hypotheses that are eventually useful for empirical neuroeconomics.
2. BASIC BUILDING BLOCKS
2.1 Environment
Let Ω be a potentially large but finite space of states of nature. These states provide some description of the environment such as which game is to be played. The states of nature are drawn randomly and independently according to some probability distribution μ ∈ Δ(Ω), where Δ(Ω) denotes the set of probability measures on Ω.
There is a finite game defined by a finite set of players I = {1,. . .,m}, for each player i a finite set of actions Ai, and for each player i a fitness function u i: ×i ϵ IΔ (A i) × Ω → ℝ, where Δ(Ai) denotes the set of probability distributions on Ai (i.e. mixed actions). Let us denote a i ∈ Ai an action of player i and a− i ∈ A − i : = ×j ∈ I ∖{i}A j a profile of actions of player i's opponents. Similarly, let αi ∈ Δ (A i) denote a mixed action of player i and α− i ∈ ×j ∈ I∖{i} Δ(A j) a profile of mixed actions of player i's opponents. We restrict the analysis to symmetric games, i.e. Ai = A for all i ∈ I and for all ω ∈ Ω, u i (αi, α− i, ω) = u f(i) (αf(i), α− f(i), ω) for all bijections f : I → I.
The framework is interpreted as follows: In each period, a state ω ∈ Ω is drawn according to the probability distribution μ on Ω. The state ω determines a symmetric finite strategic game G(ω):= 〈I,A,(u i (ω))i ∈ I〉. That is, we assume that players may not just play one game in their life but at each period games are selected according to some exogenously fixed probability distribution μ.Footnote 5 We call (Ω, μ) the environment. We say a game G(ω) is relevant if μ({ω}) > 0.
2.2 Nervous system
Each player i ∈ I has a potentially large but finite set of neurons, N i = {1,. . ., n i}. Let ⨞i be a binary relation defined on Ni for player i called “synapse”. We interpret j ⨞i j′ as “for player i neuron j projects to neuron j′”, j, j′ ∈ Ni. Since such a synapse is directed, we let ⨞i be irreflexive (but it may not be transitive or complete). If j ⨞i j′, then we call j the presynaptic neuron and j′ the postsynaptic neuron for player i. Clearly, our model of the neuron abstracts from many interesting features (see Gazzaniga et al. Reference Gazzaniga, Ivry and Mangun2002: Chapter 2).
There are special neurons called receptors used to obtain signals from the environment. Examples are the photoreceptor cells of the retina (see Gazzaniga et al. Reference Gazzaniga, Ivry and Mangun2002: Chapter 5). Perhaps one way of featuring receptors in Ni would be to require that if j ∈ Ni is a receptor then there is no j′ ∈ Ni such that j′ ⨞ij. That is, a receptor is a neuron which may project to other neurons but to which no other neuron projects to. Yet, this feature will not play a role in this note.
A neuron sequence for player i is j 0, j 1, . . . 
with j l ⨞ij l + 1 for l ∈ {0,1, . . ., − 1}. There can be loops. We call Ni = 〈N i, ⨞i〉 the anatomy of player i's nervous system or bluntly player i's “brain”. One may imagine it as a directed graph or network. The conception of the nervous system as a network has a long tradition in neuroscience that can be traced back at least to Exner (Reference Exner1894) and more recently to artificial neural networks. We will not use a neural networks approach here but just stick to primitive features of networks.
A sensory correspondence s i :Ω → 2N i for player i maps states of nature to neuronal responses thought of as neural “firing” or activation of a subset of neurons. We may want to impose conditions reflecting the constraints of the neural activity by the anatomy of the nervous system. To this extend, define for a brain Ni, a particular set of subsets denoted 
i ⊆ 2N i by N′ ∈ i if for all j ∈ N′ there exists a neuron sequence j 0,⋖, j ∈ N′ with j 0 being a receptor. We explicitly let ∅ ∈ i. We may think of an element of i as a subset of neurons that is accessible by a receptor, i.e. a “module” (Glimcher, Reference Glimcher2003: 150) or “neural circuit” accessible by a receptor. We let the sensory correspondence s i be constrained by the anatomy of the brain by imposing the condition s i(ω) ∈ i for all ω ∈ Ω. If for ω ∈ Ω the subset of neurons si(ω) is nonempty, then it must contain a receptor. Hence it can be activated by an environmental stimulus. If si(ω) = ∅ for some ω ∈ Ω, then the stimulus ω does not activate any neurons.
To complete the model, we introduce a behavioural function bi : 2N i → Δ (A) for player i that maps neural activity to mixtures over actions. An example is the activation of motor structures inducing responses of what are called effectors such as arms, hands etc. (see Gazzaniga et al. Reference Gazzaniga, Ivry and Mangun2002; Chapter 11). Note that since ∅ ∈ 2Ni, bi defines a default behaviour if no neurons are activated. Note further that since bi maps neural responses to mixtures of actions, we allow for randomness of behaviour. For instance, trichoplax adhaerens, a tiny marine animal, has no neurons (Schierwater Reference Schierwater2005). Hence, its behaviour is not controlled by a brain. Still it displays variability in behaviour that we may view here as random.
3. A DIGRESSION: NEUROECONOMICS VS. ECONOMICS
Functional neuroimaging may be viewed as mainly occupied with the description of si and bi. That is, a subject i is exposed to some stimulus ω ∈ Ω, observations of brain activity si(ω) are made through MEG, EGG, PET or fMRI (for a discussion of those methods see Gazzaniga et al. Reference Gazzaniga, Ivry and Mangun2002; Chapter 4) and a response in behaviour bi(si(ω)) is recorded. The implementation of such experiments is not as straightforward as it sounds here. To appreciate the difficulties involved, one needs to consider that the equipment requires large fixed costs. Moreover, the small sample sizes used in neuroeconomic experiments seem to suggest that the variable costs of experiments must be extremely high too. The experimental designs must meet additional challenges from potential confounding effects involved with brain scanners. Finally, typical neuroeconomic papers reveal that the data transformations and statistical analysis including their underlying assumptions are apparently difficult to report in a transparent manner.
Imaging studies of the brain yielded some empirical restrictions on si. For example let Ni = 〈 N i, ⨞i〉 be a brain. The condition s i (ω) ≠ N i for all ω ∈ Ω would capture a weak version of the Principle of Functional Segregation: No functions of the brain are performed by the brain as a whole. Similarly, the condition if s i (ω) = E ≠ ∅ then E = F′ ⋃ F″ with F′ ≠ F″ and nonempty F′, F″ ∈ i would capture a weak version of the Principle of Functional Integration: No function is performed by a single “module” of the brain alone. For a discussion of those principles, see Cohen and Tong (Reference Cohen and Tong2001).
Economics essentially follows a traditional behavioural paradigm and focuses in our game theoretic context on the optimality of strategies under complete or incomplete information. Complete information refers to the case where the player can perfectly observe the state of nature. In our framework, it would correspond to si being one-to-one or injective: for any ω, ω′ ∈ Ω, ω ≠ω′ implies si(ω) ≠ si(ω′). Incomplete information refers to the case where a player can not discriminate between some states of nature. That is, we do not rule out that for some ω, ω′ ∈ Ω with ω ≠ ω′ we have s i(ω) = s i(ω′).
Under complete information, a strategy is simply a map σi : Ω → Δ(A). It assigns to each state of nature a mixture of actions. Under incomplete information, we need to restrict explicitly the strategies to private information. In our context it means that we need to constrain it by values of the sensory correspondence (analogous to the constraining strategies to types in games with incomplete information). That is, a strategy under incomplete information is a map σi : Ω → Δ(A) subject to for any ω, ω′ ∈ Ω with si(ω) = si(ω′) implies σi(ω) = σi(ω′).
The name “strategy” may be misleading here because it suggests that σi is the object of conscious choice by player i. Since we assume a large number of states and at each period a random selection of states according to some probability distribution, such interpretation may not be appropriate in a descriptive sense. Rather, we may view a player as “programmed” to a heuristic or a rule (see Gigerenzer et al. Reference Gigerenzer and Todd2000) that is then calibrated by an evolutionary learning process as outlined in section 6. While this “programming” perspective may not be the standard interpretation in economics, it is familiar to the economists from evolutionary game theory (see Weibull Reference Weibull1995).
Note that we allow for framing: Let ω, ω′ ∈ Ω be such that ω ≠ ω′ and G(ω) = G(ω′). That is, games at ω and ω′ are formally identical but they may differ in their “colour” or “smell”. Yet, we allow the values of the feasible strategy to differ between the states. For example, we allow that administering subjects oxytocin before playing trust games as in Kosfeld et al. (Reference Kosfeld, Heinrichs, Zak, Fischbacher and Fehr2005) or Zak et al. (Reference Zak, Kurzban and Matzner2005) may alter the actions of the subjects as compared to a placebo.Footnote 6
No matter whether we focus on complete or incomplete information, in our context we may view a strategy as a composition of the sensory correspondence and the behavioural function, σi = b i ○ s i. So an analogy between neuroeconomics and economics should become clear: When economics studies informational constraints on choice behaviour, neuroeconomics studies neurobiological constraints on choice behavior by adding the focus on how the nervous system constraints information processing. Which approach one should take depends largely on the type of question one wishes to ask. If one wants to study for instance the impact of brain lesions on behaviour (a question taken up in section 5), the standard economic approach does not suffice but a model on how the nervous system constrains information processing has to be added.
4. “WELL FUNCTIONING” BRAINS
Glimcher (Reference Glimcher2003: 167) writes “(t)he goal of the nervous system is to maximize the inclusive fitness of the organism”. If a nervous system would play a strategy that is strictly dominated in the “average game of life”, then clearly it would not maximize its fitness. Therefore we assume that in our context “functioning well” shall mean to play strategies that are not strictly dominated in the “average game of life”. In experiments we usually judge a player's performance only in one isolated controlled game at a time but do not observe the player's performance in the “average game of life”. Hence, we consider as a second criterion that “functioning well” refers to the ability of choosing in all relevant situations actions that are not strictly dominated.
More formally, an action ai ∈ A is strictly dominated in the game G(ω) if there is a mixed action αi ∈ Δ(A) such thatFootnote 7u i (a i, a − i, ω) < u i (αi, a − i, ω) for all a −i ∈ A −i.
For ω ∈ Ω, let Dω be the set of actions that are not strictly dominated in the game G(ω). Define for Ω′ ⊆ Ω, a set D Ω′ ⊆ A by (i) for all ω ∈ Ω′ there exists a i ∈ D Ω′ with ai ∈ Dω, and (ii) there is no D ⫋ D Ω′ for which (i) holds. Condition (i) ensures that for each state ω ∈ Ω′ there exists an action ai in D Ω′ that is not strictly dominated in G(ω). Condition (ii) requires that D Ω′ is “minimal” in the sense that there is no smaller set of actions satisfying condition (i). That is, D Ω′ is a smallest set of actions in A with the property that for each state ω ∈ Ω′ there is exactly one action in D Ω′ that is not strictly dominated in G(ω).|D Ω′ ∩ D ω| = 1 for all ω ∈ Ω′. Note that D Ω′ may not be unique.Footnote 8 For Ω′ ⊆ Ω, let 
Ω′ denote the set of all sets of actions satisfying (i) and (ii). Note further that since Ω is finite, we must have that every D ∈ Ω′ is finite for every Ω′ ⊆ Ω. In fact |D| ≤ |Ω′| for all D ∈ Ω′ and all Ω′ ⊆ Ω.
We define the variability of the environment (Ω, μ) by ϵ (Ω,μ):= 
, where supp μ : = {ω ∈ Ω :μ({ω}) > 0} is the support of μ. Intuitively, ϵ (Ω, μ) is the minimal number of actions required that enables the play of an action that is not strictly dominated in any relevant state. By definition, ϵ (Ω, μ) ≤ |Ω |. That is, the number of states of the environment provide an upper bound on the variability of the environment. Note that the definition of the variability of the environment depends on the choice of the solution concept (here actions that are not strictly dominated) and hence on the fitness “goal”.
Let Si(Ω, Ni) be the set of all sensory correspondences from Ω to i. Similarly, let Bi(Ni, A) be the set of all behavioural functions from 2N i to Δ(A). A strategy σi :Ω → Δ (A) is feasible for the brain Ni if σi = b i ○ s i with s i ∈ S i (Ω, Ni) and b i ∈ B i (Ni, A). As mentioned in section 3, we don't view here a strategy as an object of conscious choice by the brain but rather as a heuristic or rule to which a player is “programmed”.
We define the size of the brain by β (Ni): = |i |. Note that the size is not necessarily increasing in the number of neurons but such increase requires also appropriate synapses and the connectivity to receptors and effectors. The larger the size of the brain, the more variability in behaviour it may generate and the better it can gather information about the environment. The following example is used to motivate the above definition:
Example 1. Consider the brain Ni = {1}.
i= {∅, {1}}. Hence β(Ni) = 2. The environment is given by Ω = {ω1, ω2, ω3}, μ ({ω}) > 0 for all ω ∈ Ω and a single person game as follows:
The table shows also one possible assignment of the sensory correspondence si and behavioural function bi. There are no possible assignments of si and bi that would allow the individual to choose her most preferred action in each state. The reason is simply that the size of the brain is not large enough given the variability of the environment, ϵ (Ω, μ) = 3. One more neuron would be sufficient to solve the problem.
This observation can be generalized to characterize brains that “function well” in all relevant situations.
Remark 1.The size of the brain Ni is strictly lower than the variability of the environment (Ω, μ) if and only if for any feasible strategy σi of the brain Ni there exists a relevant game G(ω) for which σi prescribes a strictly dominated action.
Proof. “⇒”: Suppose to the contrary that there exists a strategy σi feasible for Ni such that σi(ω) is not strictly dominated for all ω ∈ Ω with μ({ω}) > 0. Then |range σi | ≥ ϵ (Ω, μ). Since σi is feasible, σi ∈ b i ○ s i with s i ∈ S i (Ω, Ni) and b i ∈ B i (N i, A). Thus |range σi | = |range b i | ≤ β (N i), a contradiction to β (N i) < ϵ (Ω, μ).
“⇐”: Suppose to the contrary that β (N i) ≥ ϵ (Ω, μ). Then construct a strategy σi such that for each ω ∈ Ω with μ ({ω}) > 0, σi(ω) is not strictly dominated in G(ω). Such a strategy is feasible for Ni, a contradiction.
We denote by Σ(Ni) a finite set of strategies feasible for the brain Ni. Moreover, in light of Remark 1 we assume that if the size of the brain Ni is at least as large as the variability of the environment (Ω, μ) then Σ(Ni) contains a strategy prescribing for each relevant game G(ω) an action that is not strictly dominated. Finally, we assume that if N i = N j then Σ(Ni) = Σ(Nj).
A brain may be well adapted to its environment in the sense of not playing a strictly dominated action in any relevant situation. Yet, such a strategy may be strictly dominated by another strategy in the overall “average game of life”. Let U i(σ) := ∑ω ∈ Ω μ({ω}) ui(σ(ω), ω) denote the expected fitness of player i from playing strategy σi when opponents play σ −i (i.e. expected over the entire life for a fixed strategy profile). This is the payoff function in the “average game of life” denoted by Γ defined for a given environment (Ω, μ), the set of players I, a given profile of brains (Ni)i ∈I and for each player i/brain Ni a set of feasible strategies Σ(Ni).
A feasible strategy σi ∈ Σ(Ni) is strictly dominated by a mixture of feasible strategies ρi ∈ Δ(Σ(Ni)) in Γ ifFootnote 9u i (σi, σ− i) < u i (ρi, σ− i) for all σ− i ∈ ×j ∈ I∖ {i} Σ (N j). Note that according to this definition, a strategy of player i may become strictly dominated if player i's size of the brain increases or the sizes of her opponents’ brains increase. That is, a player with a previously “well functioning” brain may find it impossible to adapt herself well after opponents evolve more sophisticated brains.
Remark 2. Suppose that for every player i ∈ I the size of her brain Ni is weakly larger than the variability of the environment. If σi ∈ Σ(Ni) is not strictly dominated in the average game of life Γ by some other strategy feasible for Ni, then σi(ω) is not strictly dominated in G(ω) for all ω ∈ Ω with μ({ω}) > 0.
Proof. Suppose by contradiction that σi ∈ Σ(Ni) is not strictly dominated in Γ but that there exist a state ω ∈ Ω such that σi(ω) is strictly dominated in G(ω). Construct a new strategy σi* that agrees with σi on all games G(ω′) with μ(ω′) > 0 where σi(ω′) is not strictly dominated in G(ω′). In any other games G(ω″) with μ(ω″) > 0 where σi(ω″) is strictly dominated in G(ω″) let σi* (ω″) strictly dominate σi(ω″). Since β(Ni) > ɛ(Ω, μ), such strategy is feasible for Ni and by assumption such strategy is contained in Σ(Ni). Note that σi* strictly dominates σ in Γ, a contradiction.
The converse is not true. A counter example can be constructed similarly to Germano (Reference Germano2007: Example 2).
5. BRAIN LESIONS
The motivation for this section is twofold: first, in neuroscience, lesion studies are common. A lesion is a damage of brain tissue possibly separating projections between neurons or destroying neurons altogether. The effect of such lesions is then studied in patients. While some brain functions are lost due to lesions, patients are often quite well calibrated to the environment. For instance, the patient N.R. who suffered from the Balint's syndrome caused by a right pariental lesion due to a stroke can not see two objects shown to him at the same time but only sees one object at a time while speech and comprehension are normal (see Gazzaniga et al. Reference Gazzaniga, Ivry and Mangun2002: 245, 292). The second purpose of this section is to define a “set of brains” that we will use in the next section on the evolution of brains.
Given a brain N i = 〈 Ni, ⨞i 〉, define a brain N′i = 〈 N i, ⨞ ′i 〉 by N′i ⊆ N i and for j, j′ ∈ N′i, j ⨞′ij′ implies j ⨞ij′ (but not necessarily vice versa). We can view N i′ as a brain obtained from Ni by a lesion. By definition, β (N i′) ≤ β (N i). That is, the size of the brain without the lesion is weakly higher than the size of the brain with the lesion. Naturally, we assume Σ(N i′) ⊆ Σ (N i). A brain with a lesion has a lower number of feasible strategies available than the brain without the lesion.
Let 
i denote the (partially ordered) set of all brains that can be obtained from Ni by lesions. We call i the set of brains derived from Ni. In the next section, we do not necessarily interpret a brain N i′ ∈ as a brain obtained from N by a lesion. Rather, i is just a set of brains with weakly lower size than the size of brain Ni.Footnote 10
Do lesions always matter? The following example illustrates that this depends on the kind of lesion of the player's brain and the environment.
Example 2. Consider a brain given by Ni = {1, 2, 3} with 1 ⨞i 2, 1 ⨞i 3 and 2 ⨞i 3. Thus
i = {∅, {1}, {1, 2}, {1, 3}, {1, 2, 3}} and β(Ni) = 5. Consider now a lesion that severs the synapse between 2 and 3. Note that Si and β(Ni) remains unchanged. Hence such a lesion won't affect information processing and behaviour no matter how rich the environment is. Consider now a lesion that severs the synapse between 1 and 2. The size of the brain is now reduced to 3 even though no neuron was removed. Despite this “brain damage”, the player still can “function well” in below environment Ω = {ω 1, ω 2, ω 3} with μ({ω}) > 0 for all ω ∈ Ω because it is feasible for her to be programmed to si and bi given in the table:
For such an environment, the lesion won't affect her ability to play actions that are not strictly dominated in each state. This holds true even if the lesion would remove either neuron 2 or 3 altogether.
Lesions may have an externality on other players (as caretakers of patients sometimes note).
Example 3. Let the environment consist of two states, Ω = {ω 1, ω 2}, with μ({ω}) = 
for all ω ∈ Ω. The game at each state is given by the following payoff matrices:
Let both players' brains be given by N 1 = N 2 = {1}. Such a brain enables each player to play a feasible strategy given by
that selects the strict dominant action and the Pareto efficient outcome in each game. That is, for player 1 we let s 1 (ω1) = ∅, b 1 (∅) = up, s 1(ω 2) = {1}, b 1({1}) = down and analogously for player 2. Suppose now that player 1 suffers a brain lesion such that her brain with the lesion is N′1 = ∅. The above strategy is not feasible any more for player 1 with the brain damage. Only constant strategies are feasible that prescribe either up or down or a constant mixture thereof at both states. Since player 2 sticks to his dominant strategy, in expectations any constant strategy yields a fitness of 9 to player 1. Yet, player 2 incurs a much bigger fitness loss since she receives in expectations only 5. While player 1 suffered the brain damage, the healthy player 2 incurs most of the costs.Footnote 11
6. DEVELOPMENT AND EVOLUTION OF BRAIN FUNCTIONS
We are not born with a fully developed brain. For instance, in newborns the optic nerves are not developed completely but reach typical adult patterns only at the age of about 2 years. But even the nervous systems of adults maintain some neural plasticity as indicated by learning of new skills or the development of phantom sensation of amputees (see Gazzaniga et al. Reference Gazzaniga, Ivry and Mangun2002: Chapter 15). More generally, if the nervous system regulates the interaction of the organism with other organisms and the complex changing environment, there should be an evolutionary selection of nervous systems. First, we will try to analyse the question whether “successful” brain functions si and bi can develop among interacting brains in a changing environment. Second, we focus on the evolution of brains.
6.1. Development
Starting from an initial distribution of feasible strategies 
= (ρi, ρ− i) ∈ ×j ∈ I Δ (Σ (N j)) among brains, we assume that brains develop feasible strategies according to a discrete-time stochastic aggregate log-monotone dynamics defined by
where λi : ×j ∈ I Δ (Σ (N j)) → ℝ+ is a positive continuous function bounded away from zero. This dynamics is just one learning dynamics reflecting the “law of effect”: The probability of playing a certain strategy increases in the relative performance of the strategy in randomly drawn games among brains. Note that the propensity to use a certain strategy is updated with respect to randomly drawn games (instead of the average game of life). This dynamics has been studied by Cabrales and Sobel (Reference Cabrales and Sobel1992) in a standard evolutionary game setting and for our stochastic environments by Germano (Reference Germano2007).
Proposition 1. Fix an environment (Ω, μ) and a profile of brains (N j)j ∈ I. Let σi ∈ Σ (N i) be a feasible strategy of the brain Ni for some player i, which is strictly dominated in the average game Γ. If for every player j ∈ I there is initially positive probability that Nj uses any feasible strategy in Σ(Nj), then the brain Ni develops to use σi with zero probability almost surely.
Suppose further that for every player i ∈ I the size of the brain Ni is weakly larger than the variability of the environment (Ω, μ). If for some player i ∈ I, σi ∈ Σ (N i) is a feasible strategy for Ni that prescribes a strictly dominated action in some relevant game and for every player j ∈ I there is initially positive probability that Nj uses any feasible strategy in Σ(Nj), then the brain Ni develops to use σi with zero probability almost surely.
Proof. The first conclusion is a reinterpretation of Germano (Reference Germano2007: Proposition 1). The second conclusion follows from the first conclusion using Remark 2.
Since the statement is for a fixed profile of brains, the interpretation is restricted to learning and development of brains. In light of Proposition 1 it would be interesting to study the correlation between behavioural changes and the development of nervous systems in children. For instance, Harbaugh, Krause and Berry (Reference Harbaugh, Krause and Berry2001) examine to which extent consumption choices by 7- and 11-year-old children and college undergraduates satisfy the axioms of revealed preference. They find that choices by even the 7-year-olds are considerably more likely to obey revealed preference axioms than would be true if they were choosing randomly. Eleven-year-olds do better still, while college students do no better than 11-year-old children. They argue that this evidence suggests that the ability to choose rationally is not innate, but that it does develop quickly.
6.2. Evolution
Now we turn our attention to the evolution of brains. Consider a sufficiently large population of players. Each player is endowed with a brain N ∈ 
, where is a set of brains derived from some brain 
as discussed in section 5. We assume that the size of is weakly larger then the variability of a fixed environment (Ω, μ), β () ≥ ϵ (Ω, μ). We denote by η ∈ Δ () the distribution of brains within the population. For example, η (N) denotes the fraction of the population endowed with the brain N ∈ .
At each period t players are randomly and anonymously matched to play the game at ωt. If a player's brain is N ∈ , then he is programmed to some feasible strategy σ ∈ Σ (N). Let ρN ∈ Δ (Σ (N)) be the distribution of strategies in the population of players endowed with brain N. For example, ρN (σ) is the fraction of players programmed to σ ∈ Δ (Σ (N)) among all players in the population with brain N. (If η(N) = 0, then ρ N can be arbitrary.) We define ρ ∈ Δ (Σ ()) by 
. This is the fraction of the entire population programmed to σ.
We assume that the evolutionary selection of strategies within the entire population follows equation (1), i.e.
![\[
\rho^{t+1}(\sigma) =
\frac{\rho^t(\sigma) e^{\lambda(\bar{\rho}^t)(u_i(\sigma(\omega^t),
\stackrel{m-1}{\rho^t, ..., \rho^t}, \omega^t) -
u_i(\bar{\rho}^t,\omega^t))}}{\sum_{\sigma' \in \Sigma(\mathbf{N})}
\rho^t(\sigma') e^{\lambda(\bar{\rho}^t)(u_i(\sigma'(\omega^t),
\stackrel{m-1}{
\rho^t, ..., \rho^t}, \omega^t) - u_i(\bar{\rho}^t, \omega^t))}}.
\]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160331084311287-0895:S0266267108002113_eqnU5.gif?pub-status=live)
This equation may be viewed as a discrete-time version of the replicator dynamics used in standard evolutionary game theory (see Cabrales and Sobel Reference Cabrales and Sobel1992). By Remark 1, the evolutionary selection of strategies has implications on the evolution of brains:Footnote 12 If ρ (σ) = 0 for all feasible strategies σ ∈ Σ (N) of the brain N, then η(N) = 0.
Corollary 1. Given the environment (Ω, μ), consider the set of brains derived from a brain whose size is weakly larger than the variability of the environment. If initially there is a completely mixed distribution of brains η ∈ Δ () in the population of players and for each brain any feasible strategy has initially strict positive probability in the population, then evolution lets the fraction of players with a brain of strictly smaller size than the variability of environment go to zero almost surely.
Proof. For all brains N with β(N) < ɛ(Ω, μ) it follows from Remark 1 that any feasible strategy σ ∈ Σ (N) must prescribe a strictly dominated action σ (ω) for some game G(ω) with μ({ω}) > 0. Then the result follows from Proposition 1. That is, the result is just a reinterpretation of Germano (Reference Germano2007: Proposition 1).
Empirically, there is quite some variation of the number of neurons (a proxy for our measure of brain size) in organisms. For instance, trichoplax adhaerens, a tiny marine animal, has no neurons at all (Schierwater, Reference Schierwater2005) whereas human beings are estimated to have about 95 billion neurons and about 100 trillion synapses. While humans do not have the largest brain both in terms of relative or absolute volume or weight or the total number of neurons, they have the highest number of cortical neurons (for a survey see Roth and Dicke Reference Roth and Dicke2005). The cerebral cortex is often associated with “thinking”, “perceiving”, “producing” and “understanding” language but it is also involved in more basic functions such as vision, hearing, touch, movement and smell (Gazzaniga et al. Reference Gazzaniga, Ivry and Mangun2002: 70). It is the most recent structure in the history of brain evolution. The following table provides a comparison of numbers of cortical neurons in some mammals (see Haug, Reference Haug1987; Roth and Dicke Reference Roth and Dicke2005):
In light of Corollary 1, it would be an interesting empirical exercise to investigate beside the brain sizes of organisms also a measure of the variability of their environment, and check for a correlation. Note however that this does not provide a test for the result because it could well be that organisms 1 and 2 are such that the brain size of 1 is smaller than the brain size of 2 and the variability of 1's environment is higher than the variability of 2's environment but both organisms’ brain sizes are larger than their respective environment's variability.
6.3. Endogenous changes of environments
Today there are signs that human behaviour changes the environment more and more. For instance, the industrial revolution may cause global warming. Even in more primitive societies, actions today impact the environment tomorrow; for example hunters need to move on once animals are hunted, nomads need to move depending on the grazing activity of their livestock, wars destroy potentials of future production etc. Do the conclusions of the previous section remain intact in such a more realistic setting?
More formally, in contrast to section 2.1 suppose now that at each period players can influence interactively through their actions the probability with which the next state is drawn. In particular, we assume that μ (ωt + 1 |ωt, a) is the transition probability that the state is ωt+1 in period t + 1 given that the state in t is ωt and the players’ profile of actions at t is a = (a l,⋖, am). Essentially, these transition probabilities together with games {G(ω)}ω ∈Ω render the environment into a stochastic game. Analogous to the theory of stochastic games, we call player i's strategy σi Markov if at any period of time it just depends on the current state.
Example 4 (apokalupsis eschaton). There are two players. Their environment consists of two states Ω = {ω 1, ω 2}. In any of those states, either player can take either of two actions. The payoffs in each state are given by the payoff matrices. The transition probabilities associated with each state and each profile of actions are given below the payoff matrices. (Each component of the matrix corresponds to the state and action profiles above, assigning the probability of transiting to ω 1 and ω 2 respectively. We let ϵ > 0.)
![\[
\begin{array}{@{}c@{\qquad}c@{}}
{\omega _1} & \qquad {\omega _2} \\
{\left({\begin{array}{@{}c@{\quad}c@{}}
{3,3} & {0,4} \\
{4,0} & {2,2} \\
\end{array}} \right)} &\qquad {\left({\begin{array}{@{}c@{\quad}c@{}}
{- 10, - 10} & {- 10, - 10} \\
{- 10, - 10} & {- 10, - 10} \\
\end{array}} \right)} \\\\
{\left({\begin{array}{@{}c@{\quad}c@{}}
{(1,0)} & {(1 - \epsilon, \epsilon)} \\
{(1 - \epsilon, \epsilon)} & {(1 - \epsilon, \epsilon)} \\
\end{array}} \right)} & \qquad {\left({\begin{array}{@{}c@{\quad}c@{}}
{(0,1)} & {(0,1)} \\
{(0,1)} & {(0,1)} \\
\end{array}} \right)} \\
\end{array}\]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160331084311287-0895:S0266267108002113_eqnU6.gif?pub-status=live)
G(ω 1) is a standard Prisoner's dilemma with down and right being strictly dominant. In G(ω 2) any action is not strictly dominated.
We assume that the initial state is ω 1. No matter whether players have a brain or not, the dynamics in equation (1) should lead players to play the strictly dominant action in G(ω 1) starting from a completely mixed action profile. Such play leads at some point to the absorbing game G(ω 2) with very low fitness to both players. Yet, playing the strictly dominated action in G(ω 1) is part of the strategy that is strictly dominant in the average game. So there is no way in which brains as modelled in this note can develop to “function well” in the “average game of life” with the dynamics in equation (1) because “functioning well” would mean to avoid the “bad life” in game G(ω 2) altogether. Note that we could slightly perturb the payoffs and transition probabilities and the same conclusion would obtain. Thus such class of games is not negligible.
What to make of it? On one hand, we can dismiss adaptive play given by equation (1) as extremely mechanistic and backward looking and our model of the “brain” as a meaningless caricature. What would a model of a brain need to look like that is able to generate foresight required to “function well” in problems like Example 4? What enables the imagination of consequences without having to experience similar consequences beforehand? On the other hand, stories like the one of Adam and Eve show that we may (even religiously) believe that humans are created in such a way that they fail to envision consequences of their actions (despite being told about them beforehand).
In the exogenously changing environments studied in the previous section, larger brains have an evolutionary advantage in more complex environments. When the change of environment depends endogenously on the players’ actions, a larger brain can also generate more variability in behaviour and hence make the environment more variable as well. Therefore, it is not clear any more, whether larger brains maintain an evolutionary advantage over smaller brains in endogenously changing environments. It is possible to build more sophisticated examples where only the presence of a large brain in a population of “no-brainers” triggers the transition to “bad” absorbing sets of games. It is also possible to construct examples, where large brains are needed to enter relatively small absorbing sets of games and then once entered evolutionary drift reduces the brain size in the population over time because the evolutionary selection pressure is not present anymore in the small absorbing set of games.
7. SOME FURTHER DISCUSSION
What is really the relevance of such an evolutionary model? It gives a preliminary answer to the “major criticism that Marr's approach has faced”. Namely, that “it has been unclear whether evolution can be conceived of as a process that structures nervous systems to accomplish goals with enough efficiency to make the computational goal a useful starting point for neurobiological analysis,” (Glimcher Reference Glimcher2003: 166). It does shed some light on the dependence of the appropriate brain size on the variability of the environment but such a relationship is far from surprising and the model falls short of generating a hypothesis that is really testable. It does also question the ability of evolution and development to adapt appropriately to an environment that can be changed by the players themselves. But given the crude model of the nervous system and the evolutionary process, how seriously should it be taken?
One important aspect from an economic point of view – which is not considered here at all – is that large brains in humans constitute large investments. This large investment does not only come in form of bodily capital (the extreme rapid growth requires prenatally about 60% of the metabolism, see Roth and Dicke Reference Roth and Dicke2005: 254) but large brains also demand education and hence further investment by society into human capital. Moreover, the “maintenance” of such large brains consumes about 20% of the total metabolism while it constitutes only 2% of the body weight (Roth and Dicke Reference Roth and Dicke2005: 254). A more comprehensive theory of the development and evolution of the brain needs to take into account the trade off between the higher costs of a larger brain and the more sophisticated behaviour it may generate. Robson and Kaplan (Reference Robson and Kaplan2003) present such a “brain-capital” theory.
This note uses results by Germano (Reference Germano2007) and he may have anticipated such use when he wrote (p. 324) “(I)t seems that some of the main challenges lie in the characterizing ‘good’ rules that ideally apply to a wide range of games and environments, and linking them to actual cognitive (or genetic) behaviour”. He also presents additional results such as on the elimination of strategies that are not rationalizable, Nash equilibria in the average game of life as limit points under convergence etc. It would be interesting to consider such strategically more sophisticated solution concepts because Dunbar and Shultz (Reference Dunbar and Shultz2007) suggest that the strategic demand from living in complex societies selected for sophisticated brains whereas our focus on actions/strategies that are not strictly dominated covers mainly the demands upon the brain made by the ecological variability.
Our model has nothing to say about internal mental conflicts modelled in recent papers on neuroeconomic theory by Benhabib and Bisin (Reference Benhabib and Bisin2005), Bernheim and Rangel (Reference Bernheim and Rangel2004), Brocas and Carrillo (Reference Brocas and Carillo2008a, Reference Brocas and Carillo2008b) and Fudenberg and Levine (Reference Fudenberg and Levine2006). Our hope is that a more sophisticated evolutionary approach could shed some light on the evolution of multiple selves. A first attempt is presented by Livnat and Pippenger (Reference Livnat and Pippenger2006) who show what computational constraints give optimally rise to “multiple selves”. However, they do not model the evolutionary selection of players with multiple selves in the spirit of evolutionary game theory.
