Distinguishable Decision Problems

We present a few examples that illustrate behavior that would be impossible to ratio- nalize in a standard individual decision framework. In all these examples we assume, for

simplicity, thatAand P are …nite sets and (a)is the identity map. The preferences of the decision maker are represented by an utility functionu:A P _{! <}. We distinguish between pure and random behavioral decisions. Let (^a) = arg maxa2Au(a;^a). Apure

action behavioral equilibriumis an action pro…lea such thata ₂ (a ). Let (A)

denote the set of probability distributions over the set of actions. A random strategy is

2 (A), where (a) is the probability attached to action a. A random distribution over the set of psychological states is ₂ (A), where (^a) is the probability attached to psychological state ^a. A random decision state is a pair ( ; ). Given a random decision state( ; ), the payo¤ to the decision maker is

w( ; ) =X

a2A X p2P

(a) (^a)u(a;^a) (2.16) A consistent random decision state is a pair( ; )where = . Arandom behavioral

equilibrium is a pro…le such that ₂arg max ₂ (A)w( ; )17.

In each example, the decision problem is represented by a payo¤ table where rows are actions and columns are the psychological states. Under the assumptions made so far, consistent decision states are the diagonal of these payo¤ tables.

Example 4 A unique ine¢ cient behavioral decision in dominant actions: addiction

Consider the following payo¤ table:

p1 p2

a1 1 1

a2 2 0

(2.17)

17_{I acknowledge feedback from the PhD examiners on the fact that this de…nition of random be-}

havioural equilibrium assumes stochastic independence between actions and psychological states, in- validating the feedback e¤ect. A perfect correlated equilibria might have been a better approach to undertake. In any case, a random behavioural equilibrium has been de…ned to introduce some examples as illustrations but the main results of this thesis are not a¤ected by this de…nition.

We interpret these payo¤s as an example of addiction where a2 corresponds to smoking

and a1 corresponds to not smoking and pi to di¤erent health states of the individual

(p2 is less healthy than p1). In this case, in a behavioral decision problem, the decision

maker always choosesa2asa2 is the dominant action for each value ofp: if the individual

takes her health state p as given she always prefers to smoke. The unique behavioral decision outcome is(a2; p2)with a payo¤ of0. However, note that the consistent decision

state (a1; p1) with a payo¤ of1 is the only element of M: once the individual takes the

feedback from actions to health states into account, she always chooses not to smoke

Example 5 No pure action behavioral decision: the grass is always greener on the other

side

p1 p2

a1 0 1

a2 1 0

(2.18)

We interpret these payo¤s as an example of a situation where the individual makes a choice between two di¤erent lifestyle so thatpi denotes a speci…c lifestyle and ai denotes

the action that chooses location pi. Starting from p1, the decision-maker prefers a2 to

a1 while starting fromp2, the decision-maker prefersa1 toa2: the individual always be-

lieves that the grass is greener on the other side. There is no behavioral decision in pure strategies. The decision-maker is, however, indi¤erent between both the two consistent decision-states(a1; p1) and (a2; p2).

This example demonstrates that, in general,E may be empty even when M isn’t. How- ever, given the discussion so far, a behavioral decision outcome can be interpreted as a Nash equilibrium of a two person game so that as long as A and Q are …nite, a mixed strategy behavioral decision outcome always exists.

Example 6 Equilibrium in weakly dominated actions and domination by random ac- tions p1 p2 p3 a1 0 0 0 a2 0 1 2 a3 0 2 1 (2.19)

In this example, there are two behavioral equilibria, one in pure actions,(a1; p1)and

the other random,(1 2a2+ 1 2a3; 1 2p2+ 1

2p3). Note that in the pure action equilibrium(a1; p1)

the decision-maker is choosing a weakly dominated action and at the random equilibrium (1₂a2+1₂a3; 1₂p2+1₂p3), the decision-maker is strictly better o¤ than at (a1; p1). Note also

that there is no pure action that (strictly) dominatesa1 although there are a continuum

of random actions qa2+ (1 q)a3, 0< q <1, that strictly dominates a1.

Example 7 Multiple welfare ranked equilibria: aspirations

p1 p2

a1 1 0

a2 0 2

(2.20)

We interpret these payo¤s as an example of an aspiration failure. Let a1=as under-

taking an action that perpetuates the status quo and a2=undertaking that changes the

status quo, withp2 ="high aspirations" and p1 ="low aspirations" being the consistent

psychological states associated with a1 and a2 respectively. In this example, there are

two strict behavioral decision outcomes (a1; p1) and (a2; p2). Note that the pure ac-

tion equilibrium (a1; p1) is dominated by the pure action equilibrium (a2; p2). When

low, (a2; p1) (a1; p1). Thus, the behavioral decision outcome (a1; p1)is an instance of

an aspirations failure.

Example 8 More information may make the decision-maker worse-o¤

Consider a decision problem with payo¤ relevant uncertainty, with two states of the world_f 1; 2gwhere the payo¤ tables are

1 ! p1 p2 p3 a1 1 0 0 a2 0 3 1₂ a3 1 4 1 (2.21) 2 ! p1 p2 p3 a1 1 4 1 a2 1₂ 3 0 a3 0 0 1 (2.22)

Suppose, to begin with, the decision-maker has to choose before uncertainty is resolved. At the time when she makes the decision, the individual attaches a probability 1₂ to 1

and 1₂ to 2. In this case, expected payo¤ matrix is

p1 p2 p3

a1 0 2 1₂

a2 1₄ 3 1₄

a3 1₂ 2 0

(2.23)

It follows that the unique behavioral equilibrium is (a2; p2) with expected payo¤3.

Next, suppose that the decision-maker knows with probability one the true state of the world. Then, when the state of the world is 1, a3 strictly dominates all other

actions and the unique behavioral equilibrium is (a3; p3) with payo¤ 1 and when the

state of the world is 2,a1 strictly dominates all other actions and the unique behavioral

equilibrium is (a1; p1) with payo¤1. Therefore, the decision-maker is worse-o¤ with

more information18 19_.

Example 9 Autonomy versus non-autonomy

Consider the payo¤ table in matrix 2.18. In that example, if the decision maker took into account the feedback e¤ect from actions to the utility parameter and maximized the induced utility function v(:), v(a1) = v(a2) = 0. Therefore, a fully autonomous

decision-maker who takes into account all the consequences of her actions would obtain a payo¤ of 0. However when the decision-maker doesn’t take this feedback e¤ect into account, we have already seen that there is a unique random outcome of the behavioral decision problem(1₂a1+1₂a2;1₂p1+1₂p2)with an expected payo¤ of 1₂ >0. On the face of

it, it would seem that a non-autonomous decision-maker will be better-o¤ than a fully autonomous decision-maker. But this interpretation isn’t strictly true. In fact, if a fully autonomous decision maker is also allowed to choose mixed strategies in the payo¤ in matrix 2.18, she will also randomize _fa1; a2g by choosing the probability distribution

1 2;

1 2 .