Aspiration Traps

2.1 Model

In period t=0 an individual has to make some fundamental choice, e∈E, at a cost c(e). For a given choice e, the game of life, to be played in t=1, is described by a tuple Ge=(X,ue,B,β), where the strategy set of the individual is X, and her payoff function is ue:X×B→ ℝ. Differently from a standard decision-theoretic problem, the utility of the individual depends also on her attitude (beliefs, aspirations) b∈B. When choosing a strategy x(e,b) in the game of life so as to maximize ue, the individual takes as given her attitude b∈B (and, of course, also the fundamental choice e she has already taken at t=0). In reality, given e the attitude b is determined, jointly with x(e,b), by some preference formation mechanismβ:E→B, as follows:

When considering in retrospect a deviation e′, the individual understands correctly the effect it would have on her utility function ue′, but she takes as given her already acquired attitude β(e). ^[4] That’s why, typically, a self-justifying e does not maximize eq. [1].

We can identify conditions that guarantee the existence of a self-justifying choice.

A1. The sets E, B and X are non-empty compact convex subsets of a Euclidian space

A2. For all e∈E, b∈B the function ue(x,b) is strictly quasi-concave and continuous in x, and for all b∈B the indirect utility ue(x(e,b),b)−c(e) is continuous and quasi-concave in e.

A3. β:E→B is continuous.

The following proposition is a standard application of Kakutani’s fixed point theorem.

Related notions of equilibrium in decision problems or games have been discussed by Geanakoplos, Pearce, and Stacchetti (1989), Rabin (1993), Yariv (2005), Dalton and Ghosal (2012) and Bracha and Brown (2012).

In models with belief-dependent utilities, however, the “preference formation mechanism” implies that beliefs should be correct at equilibrium.

Here we allow for a rather general interpretation of the “preference formation mechanism” and of the notion of consistency (in the example of this section b is indeed a belief, but it need not be correct at equilibrium; in the example in the next section b represents more generally a “preference type,” and the preference formation mechanism reflects evolutionary stability in a strategic context).

For us, the distinction between “big” choices and the day-to-day choices in the game of life is crucial. The standard argument to justify the assumption of perfect foresight or rational expectations refers to the repetition of similar choices in a stable environment, and this assumption is untenable for self-transforming “big” choices. One possible interpretation is that of a self-justifying choice as a non-recurrent “big” choice that the individual made when she was still young and lacking a clear vision of the repercussions of her choice.

Formally, we have a Nash equilibrium between a mature individual and the unconscious preference-formation mechanism of the young, but the use of the Nash equilibrium notion is just a particular way to model the individual as embedded in society, in-between the two extreme models of the individual that can be encountered in the social sciences. Unlike in the standard Economic model, the individual is not atomistic, and her preferences are not well defined and do not form outside a social context. On the other hand, the individual is not a “cultural dupe,” a leaf in the swaying winds of social norms and forces.

In the proposed model, the two approaches are interwoven. Regarding “big” choices the individual is subject to the socialization process, while she is a standard economic maximizer concerning day-to-day alternatives. Embeddedness is expressed by the idea that these two dimensions are at equilibrium with one another.

2.2 Example

The game of life involves the choice of a production strategy under uncertainty. There are two possible states of nature, s={good,bad}, one of which will materialize after the individual has chosen her strategy. An individual has one unit of time to allocate between two activities, one safe and one risky. The safe activity yields one unit of consumption independently of the realized state. The risky activity yields two units of consumption if the state is s=good, and e<1 units of consumption if the state is s=bad.

We could interpret this as a very stylized representation of the following situation. The first activity represents work in the traditional sector, which guarantees a safe return of one unit. The second activity offers a prospect of a higher return if things go well, but is subject to risk. In the game of life the individual has to choose how to allocate her time between the two activities. When things do not go well, the payoff of the risky activity is indexed by e, a variable which the individual has to choose before she enters the game of life. We may interpret it as the level of education, capturing the idea that people with better education may have higher reservation utility if things go wrong in the risky activity.

For a given belief b∈[0,1] over the occurrence of the good state s=good and a given utility for consumption v(⋅), the individual chooses the share of time to spend in the risky activity, x∈[0,1], to maximize

We denote her optimal choice for given values of e and b by x(e,b).

We now have to specify the preference-formation mechanism. In this example, we adopt the one proposed by Brunnermeier and Parker (2005). Their starting point is the observation that beliefs affect the expected utility of the individual in two ways: indirectly, by influencing her choices, and directly, as weights on possible future contingencies. For example, an optimistic belief may lead to a biased choice, but its direct effect on the individual perception is an increase in well-being. Brunnermeier and Parker define optimal beliefs as those that would be unconsciously chosen to maximize the expected sum of utility as perceived today and tomorrow (see below for a formal definition in the context of our example). They interpret the mechanism as the result of social forces, like the fact that parents induce children to have a positive view of the world, or that happier individuals tend to be healthier.

In our example, if π∈[0,1] is the objective probability of s=good, the optimal belief b maximizes:

Optimal beliefs for a given level of e are denoted by β(e). The preceding expression can be interpreted as the expectation (using the objective probability) of the sum of two terms: the expected utility as perceived by the individual at the beginning of the game of life, when she chooses x, and her actual expected utility over the possible states of nature. The definition of optimal beliefs takes into account their influence on the choice of x, and therefore their consequences in terms of actual utility in each state of nature, but also their direct effect on the individual’s perception of the situation.

Before the game of life starts, the individual chooses a level of education e, at a cost c(e). She is fully aware of the nature of the game of life. In particular she knows how her choice of e today will affect the payoff of the risky activity, and she correctly anticipates her own optimal strategy x(e,b) tomorrow for a given belief b. But she is not aware of the way in which her own beliefs will evolve.

Her choice of education e∈E is self-justifying if, for all e′,

That is, when contemplating in retrospect an alternative fundamental choice e′, the individual maintains her already acquired belief β(e).

To fix ideas, let v(⋅)=log(⋅), π=0.5, E={0,0.1,0.33}, c(0)=0, c(0.1)=0.01, c(0.33)=0.11. Optimal beliefs turn out to be b(0)=0.5, b(0.1)=0.7, b(0.33)=0.78, with associated optimal choices of strategy x(0,b(0))=0, x(0.1,b(0.1))=0.47, x(0.33,b(0.33))=0.95.

One can check that both e=0 and e=0.1 are self-justifying choices. If she chooses to take some education, e=0.1, the individual ends up investing more in the risky activity, adopting more optimistic beliefs and being happier: the utilities are u0(x(0,b(0)),b(0))−c(0)=0 and u0.1(x(0.1,b(0.1)),b(0.1))−c(0.1)=0.09, respectively.

The choice e=0 is thus an aspiration trap. In a society in which e=0 is the established norm, a social planner who would force or tempt the individuals into choosing e=0.1 would be opposed by the old (with the already acquired belief b(0)=0.5), but would eventually be thanked by the young when they mature and acquire the belief b(0.1)=0.7. The choice e=0.1 would then be sustained as self-justifying, with no need for further policy intervention. With e=0.1, individuals in this society would be mildly optimistic, invest to some extent also in the risky activity and would be happier than under the previous regime of no-education e=0 and its consequential despairing realism.

In fact, an individual who would have taken an even higher level of education, e=0.33, would have invested even more in the risky activity, x(0.33,b(0.33))=0.95, become even more optimistic, b(0.33)=0.78, and happier, u0.33(x(0.33,b(0.33)),b(0.33))−c(0.33)=0.19. The choice e=0.33 is not a self-justifying choice, though being so optimistic the individual would regret, ex post, to have spent so much on education.

That’s why even an omniscient planner who cares about happiness cannot design a policy which would eventually make e=0.33 the established norm. If e=0.33 were continuously forced, individuals would live under the impression that such an intense level of education is exaggerated, and that they would be happier with less education. The impression is false, but this cannot be comprehended by the individuals, who cannot actually imagine their conception of the world and the way they would live their life had they been less educated.

Whatever the particular form of the preference formation mechanism, if the working of this mechanism cannot be assumed to be transparent to the individual our equilibrium notion is relevant, and the type of aspiration trap illustrated in the example is a possible outcome.

3.1 Model

Individuals are i=1,2,…,I. In period t=0 each individual i has to make some fundamental choice, ei∈Ei, at a cost ci(ei). For a given profile e∈E=×iEi, the game of life, to be played in t=1, is a tuple Ge=((Xi,uie,Bi,)i∈I,β). The strategy set of each individual is Xi, her set of potential attitudes is Bi and her payoff function is uie:X×Bi→ℝ, where X=×iXi. When choosing strategies in the game of life, individuals take attitudes as given. Attitudes are actually determined, jointly with individual strategies, by a preference formation mechanismβ:E→B, where B=×iBi. Definition 1 thus generalizes to:

When considering a deviation ei′, the individual understands correctly its effect on the structure of the game of life, and on the equilibrium behavior there, but she takes as given the other individuals’ fundamental choices and the attitudes β(e).

3.2 Example

The game of life consists of pairwise interactions. Individuals randomly meet to play in pairs i,j a symmetric game with material payoffs Π indexed by the levels of fundamental choices (“education”) (ei,ej)∈E⊂ℝ, to be decided beforehand:

Individuals maximize perceived payoffs

where bi∈ℝ is an unconscious attitude, and uiei(xi,xj,0)=Π(xi,xj;ei) (with no bias, bi=0, the individual’s perceived payoff coincides with her material payoff). For given levels of education and attitudes, a Nash equilibrium of the game defined by uiei(⋅,⋅,bi),ujej(⋅,⋅,bj) is

If there are multiple equilibria, we assume that across the pairwise interactions with parameters (ei,ej,bi,bj), these equilibria are played according to a distribution that assigns positive probabilities to all of them. ^[7] Equilibrium attitudes β(ei,ej)=bi∗(ei,ej),bj∗(ei,ej) are such that

where the expectation is taken over the distribution of play of Nash equilibria (A.1) with opponents that have chosen ej. We note that β(ei,ej) need not be unique, that is, β may be a correspondence.

Before the game of life starts, each individual i chooses a level of ei at a cost c(ei). Individuals understand the effects of the choice of ei on payoffs Π, but are not aware of the possible effect on the equilibrium attitudes, since they are unable to go through the mental exercise regarding the future formation of their attitudes. If ε is the level of education chosen by everybody, and β(ε,ε)=(b∗,b∗) are the associated attitudes, an individual i believes that if she were to choose ei′ while everybody else still chose ε, the corresponding equilibrium strategies in an interaction in which she is assigned role i would be

Given that everybody else is choosing ε, a level of education ε is self-justifying if

for all ei′∈Ei, for i=1,2.

To fix ideas, let the game of life be a game with material payoffs as follows:

fundamental choices, ei∈0,2 (no-education, education) affect the material payoff of cooperation, (C,C).

Education is chosen before the game of life starts, and it has a cost ci(0)=0,ci(2)=δ<0, small.

In any given pairwise interaction, individuals maximize perceived payoffs

where bi∈{0,2} (materialistic, prosocial) is an unconscious attitude. Consider first a scenario where the population playing the game of life is composed of educated individuals. With e1=e2=2, material payoffs are

This is a prisoner’s dilemma: (D,D) is the equilibrium among materialistic players.

On the other hand, when a prosocial row player meets a materialistic column player, their utilities are

Again, the unique equilibrium is (D,D), and the prosocial gets the same material payoff, 3, as does the materialistic.

Things change when two prosocial players meet. In this case their utilities are

The game has two equilibria – (C,C) and (D,D). If at some of these meetings the efficient equilibrium (C,C) is played, the prosocial attitude is dominant in the attitude game and therefore it is the unique equilibrium in that game. The expected material payoff in the game of life among educated and prosocial players is therefore larger than 3.

We conclude that in a population of educated and hence prosocial players, education is self-justifying. Indeed, fix prosocial attitudes, and consider a player contemplating the consequences of not taking education. Her payoff as a column player meeting an educated row player would be:

D would be a dominant strategy for her, leading to a payoff of 3 in the unique Nash equilibrium of this game, smaller than her payoff with education.

We now show that no-education is also a self-justifying choice. With e1=e2=0, material payoffs are

and utilities are

Whatever b1,b2∈{0,2} are, D is a dominant strategy. Prosocial and materialistic attitudes lead to the same material payoff, and any (mixture) of the two could be the outcome of the attitude selection game.

Regardless of the mixture of attitudes prevailing in the population, ei=0 by all individuals i is a self-justifying choice. Indeed, consider a player contemplating the consequences of taking education. Her payoff as a column player meeting an uneducated row player would be

Now the row player has a dominant strategy, D, whatever his attitude, and the equilibrium outcome is again (D,D). Education gives no advantage: it is not worth its cost.

Thus, the choice of no-education by everybody is an aspiration trap. Notice that in some sense, this trap is even more severe than the trap of a Pareto-inferior Nash equilibrium of a standard coordination game. Here, when everybody is materialistic, low-cost education seems unworthy to each individual even under a hypothetical scenario by which everybody would have chosen education. Even a joint move from no-education to education cannot be perceived as beneficial when individuals are unconscious of the attitude-formation mechanism.