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Bayesian–Nash equilibria in theory of planned behaviour

L. Almeida â , J. Cruz â , H. Ferreira â & A. A. Pinto â

a University of Minho , Minho, Portugal Published online: 05 Jan 2010.

To cite this article: L. Almeida , J. Cruz , H. Ferreira & A. A. Pinto (2011) Bayesian–Nash equilibria in theory of planned behaviour, Journal of Difference Equations and Applications, 17:7, 1085-1093, DOI: 10.1080/10236190902902331

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Bayesian – Nash equilibria in theory of planned behaviour

L. Almeida, J. Cruz, H. Ferreira and A.A. Pinto*

University of Minho, Minho, Portugal

(Received 18 February 2009; final version received 2 March 2009) The theory of planned behaviour studies the decision-making mechanisms of individuals.

We propose the Bayesian–Nash equilibria as one, of many, possible mechanisms of transforming human intentions in behaviour. This process corresponds to the best strategic individual decision taking in account the collective response. We show that saturation, boredom and frustration can lead to splitted strategies, in opposition to no saturation that leads to a constant strategy.

Keywords:game theory; theory of planned behaviour; decision-making AMS Subject Classification: 91A40; 91A80; 91E99

1. Introduction

Considering the application of Game Theory concepts to an individual and group model behaviour, we introduce in literature of the theory of planned behaviour or reasoned action, the Bayesian – Nash equilibria. The main goal in planned behaviour or reasoned action theories (see Ajzen [1] and Baker [4]) is to understand and forecast how individuals turn intentions into behaviours. We create a game theoretical model, inspired in the works of Cownley [7] and Wooders [5,7], where we consider individual characteristics of the individuals described as taste type and crowding type. The taste type characterizes the inner characteristics of an individual underlying their welfare (utility) function.

The crowding type of an individual characterizes his influence in the welfare function of the other individuals. We also distinguish two different worlds. The ‘platonic’ idealized psychological world, in our model, consists of well-defined individuals with a given taste and crowding types and welfare function. The ‘cave’ psychological world, in our model, consists of individuals, whose taste and crowding types follow the shadows of the idealized taste and crowding types according to a given probability distribution. In the

‘platonic’ idealized psychological world, we present necessary conditions for a given individual to adopt a certain behaviour decision according to the Nash-equilibria. We use these results to understand the behaviour decision of the individuals in the ‘cave’

psychological world. We also study how saturation, boredom and frustration can lead to splitted strategies, and no saturation situations can lead to a constant strategy. In the works of Driskel [9], Salas [8,9] and Sternberg [13] is presented a definition of leader as the individual that can influence others and experts as individuals with a well-defined behaviour and special ability to understand the behaviour of others. In [2], we present

ISSN 1023-6198 print/ISSN 1563-5120 online q2011 Taylor & Francis

DOI: 10.1080/10236190902902331 http://www.informaworld.com

*Corresponding author. Email: aapinto1@gmail.com Vol. 17, No. 7, July 2011, 1085–1093

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a possible psychological/mathematical concept of leaders and experts and we study their characteristics that have a positive or negative influence over others behavioural decisions.

2. Theory of planned behaviour or reasoned action

The theory of planned behaviour or reasoned action can be summarized in Figure 1 (see Ajzen [1]), where we can observe that external variables are divided in three categories:

intrapersonal associated to individual actions; interpersonal associated to the interaction of the individual with others and sociocultural associated to social values. This external variables influence, especially, the intermediate variables which are also sub-divided in three major items. The social norms can be the opinions, conceptions and judgments that others have about a certain behaviour (e.g. the others think I should stop smoking or I should do more exercise);

attitudes are personal opinions in favour or against a specific behaviour (e.g. I like to do exercise, it would be good to stop smoking); and self-efficacy is the extent of ability to control a certain behaviour (e.g. I can do exercise, I can stop smoking). These external and intermediate variables determine a consequent intention to adopt a certain behaviour.

3. Game theory: ‘platonic’ idealized psychological world

In our ‘platonic’ idealized psychological world, inspired in Plato’s world of thoughts or of the intelligible reality, we have no uncertainties, so we consider that the individuals are pure in the sense that the external and intermediate variables of the model are constant, instead of following a probability distribution. We can have a number of finite or infinite number of distinguishable taste and crowding types.

In this model, the individuals will choose a certain behaviour/groupg[G. Those choices will be done, taking in account their characteristics and personal preferences (taste type) and the other individuals observable characteristics (crowding club vector). The goal is to present a decision mechanism for the individuals, taking in account their and the others types.

Let us consider a finite numberSof individuals. For each individuals[S, we distinguish two types of characteristics:taste typeandcrowding type. We associate to each individual s[S onetaste typeTðsÞ ¼t[T that describes the individuals inner characteristics, not always observable by the other individuals. We also associate to each individuals[Sone crowding typeCðsÞ ¼c[Cthat describes the individuals characteristics observed by the others and that can influence the welfare of the others. We associate, in the theory of planned behaviour or reasoned action, the intrapersonal external variables and the attitude and self Figure 1. Theory of planned behaviour.

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efficacy intermediate variables to the taste type and the interpersonal and sociocultural external variables and the social norms intermediate variable to thecrowding type.

The individuals, with their own characteristics, can define a strategy G:S!G, i.e. each individuals[Schooses the behaviour/group that he would like to belongGðsÞ.

Each strategyGcorresponds to an intention in the theory of planned behaviour.

Given a behaviour/group strategyG:S!G, thecrowding vector m^GðGÞ[ðN^CÞ^Gis the vector whose components are m^gðGÞ, with g[G that define the number m_c^g of individuals ingthat have crowding typec[C, i.e.

m^g_c¼#{s[S:GðsÞ ¼g^CðsÞ ¼c}:

We denote byst;cthe individualswith taste typetand crowding typec. We measure the level of welfare, or personal satisfaction, that an individualst;cacquires by belonging to a behaviour/groupg[Gwithcrowding vector m^GðGÞ, using a utility function

u_t;c:G£ðN^CÞ^G!R ut;cðg;m^GÞ ¼V^g_t;cþf^g_t;cðm^GÞ

whereV^g_t;cmeasures the satisfaction level that each individualst;chas in belonging to a behaviour/groupg[Gandf^g_t;cðm^GÞmeasures the satisfaction level that an individualst;c

has taking in account, for each crowding typec⁰[C, the number of elementsm^g_c0 that exist in every groupg[G.

The individuals distributionG^*:S!Ginto the different groups is aNash equilibrium behaviour/group, if given the choice options of all individuals, no individual feels motivated to change his behaviour/group, i.e. its utility does not increase by changing his behaviour/group decision (see Pinto [12]).

Figure 2. Game theoretical model/theory of planned behaviour in the ‘platonic’ idealized psychological world.

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Our new dictionary between our game theoretical model and the theory of planned behaviour, in the ‘platonic’ idealized psychological world, is now summarized in Figure 2 (see Almeida [3]).

In what follows we will assume, for simplicity, thatf^g_t;c:ðN^CÞ^G!Ris linear, i.e.

f^g_t;cðm^GÞ ¼X

c⁰[C

A^g;c_t;c⁰m^g_c0;

whereA^g;ct;c⁰ evaluates the satisfaction that each individualst;chas with the presence of an individual with crowding typecin groupg.

We denote byS_ðt;cÞthe group of all individualss_t;cwith the same taste typet[Tand the same crowding typec[C. Letnðt;cÞcorrespond to the number of individuals inS_ðt;cÞ. An interesting way to interpretS_ðt;cÞis to consider thatnðt;cÞis the number of times that a same individualst;chas to take an action. In this case,A^g;c_t;c .0 can be interpreted as the individual positive reward by repeating the same group/behaviour choicec[C, i.e. the individualst;cdoes not feel a saturation effect by repeating the same choice. On the other hand,A^g;c_t;c ,0 can be interpreted as the individual negative reward by repeating the same group/behaviour choice c[C, i.e. the individual st;c feels a saturation, boredom or frustration effect by repeating the same choice.

3.1 Individuals that like to repeat the same behaviour (no-saturation)

In this section, we consider the hypothesis that A^g;c_t;c .0. We exploit situations where no-saturation can lead to a constant individual strategy.

Lemma1.LetG^*be a Nash Equilibrium. For every taste type t[T and every crowding type c[C, all the individuals s[Sðt;cÞ, with the same taste type and the same crowding type, belong to a same groupG^*ðS_ðt;cÞÞ.

Hence, considering thatnðt;cÞrepresents the number of times that a same individuals_t;c has to take an action, one concludes that the individual s_t;c always chooses the same behaviour/groupg, for every Nash Equilibrium behaviour/group.

Proof. Let us suppose that for a group strategyG:S!Gindividuals with same taste typet and same crowding typecbelong to more than one behaviour/group. Let us denote bygthe group where these individualsst;cattain the highest welfare (does not need to be unique).

Then any individualst;cthat belongs to another groupg⁰by moving into gincreases his welfare because A^g;c_t;c .0. Hence G:S!G is not a Nash Equilibrium behaviour/

group. A

We define theworst neighbours WNgðt;cÞof the individualst;cin the groupgby WNgðt;cÞ ¼V^g_t;cþ X

c⁰[C;A^g;c_t;c⁰,0

A^g;c_t;c⁰X

t⁰[T

nðt⁰;c⁰Þ;

where V^g_t;c represents the valuation of the individual s_t;c in the group g, and P

c⁰[C;A^g;c_t;c⁰,0A^g;c_t;c⁰P

t⁰[Tnðt⁰;c⁰Þ represents the worst neighbours that the individual st;c

can have ing.

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We define thebest neighbours BN_g⁰ðt;cÞof the individualst;cin the groupg⁰by BNg⁰ðt;cÞ ¼V^g_t;c⁰ þ X

c⁰[C;A^g;c_t;c⁰.0

A^g;c_t;c⁰X

t⁰[T

nðt⁰;c⁰Þ;

where V^g_t;c⁰ represents the valuation of the individual st;c in the group g⁰ and P

c⁰[C;A^g;c_t;c⁰.0A^g;c_t;c⁰P

t⁰[Tnðt⁰;c⁰Þ represents the best neighbours that the individual st;c

can have ing⁰.

Lemma2.If WNgðt;cÞ$BNg⁰ðt;cÞ, for all g⁰[Gn{g}, thenG^*ðS_t;cÞ ¼g for every Nash equilibriumG^*.

Hence, considering thatnðt;cÞrepresents the number of times that a same individuals_t;chas to take an action, one concludes that the individualst;cchooses the same behaviour/groupg, independently of the Nash Equilibrium behaviour/group considered.

Proof. It follows from the construction of the worst neighboursWNgðt;cÞand of the best

neighboursBNg⁰ðt;cÞ. A

We define

A~^g;c_t;c ¼

A^g;c_t;cnðt;cÞ if A^g;c_t;c .0 0 if A^g;c_t;c ,0 8<

: ;

whereA~^g;c_t;c represents the gain that the individuals_t;chas in being in the company of his pairs in groupg.

We define thecommunity worst neighbours CWNgðt;cÞof the individualsSðt;cÞin the groupgby

CWNgðt;cÞ ¼WNgðt;cÞ þA~^g;c_t;c We define

A~^g;c_t;c ¼

A^g;c_t;cð12nðt;cÞÞ if A^g_t;c⁰^;c.0 A^g_t;c⁰^;c if A^g_t;c⁰^;c,0 8<

: ;

whereA~^g;c_t;c represents the necessary discount (or loss) when the individualst;cis alone in groupg⁰.

We define thecommunity best neighbours CBNg⁰of the individualsSðt;cÞin the group g⁰by

CBN_g⁰ðt;cÞ ¼BN_g⁰ðt;cÞ þA~^g;c_t;c;

Lemma3.If CWN_gðt;cÞ$CBN_g⁰ðt;cÞ, for all g⁰[Gn{g}, then there is at least one Nash EquilibriumG^*such thatG^*ðS_ðt;cÞÞ ¼g.

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Hence, considering thatnðt;cÞrepresents the number of times that a same individualst;chas to take an action, one concludes that the individualst;cchooses the same behaviour/groupg, at least for one Nash Equilibrium behaviour/group.

Proof. It follows from the construction of the worst neighboursCWNgðt;cÞand of the best

neighboursCBNg⁰. A

3.2 Boredom and Frustration

In this section, we consider the hypothesis thatA^g;c_t;c ,0. We exploit the situations where boredom and frustration can lead to the splitting of the individuals decisions for different groups/behaviours.

We define theworst lonely neighbours WNlgðt;cÞof the individualst;cin the groupgby WNlgðt;cÞ ¼V^g_t;cþ X

c⁰[C;c⁰–c;A^g;c_t;c⁰,0

A^g;c_t;c⁰X

t⁰[T

nðt⁰;c⁰Þ:

Lemma 4. Let gA and gB be two groups in G. If, for every i[{A;B} and every g⁰{gA;gB},

WNg_iðt;cÞ#BNg⁰ðt;cÞ#WNlgiðt;cÞ;

for every g⁰–gi then G^*ðS_t;cÞ>gA–Y and G^*ðS_t;cÞ>gB –Y, for every Nash EquilibriumG^*.

Hence, considering thatnðt;cÞrepresents the number of times that a same individuals_t;c has to take an action, one concludes that the individual s_t;c splits its decision, at least, between the groupsg_A andg_B, independently of the Nash Equilibrium behaviour/group.

Proof. It follows from the construction of the worst lonely neighboursWNlgðt;cÞ, of the best neighboursBNg⁰ðt;cÞand of the worst neighboursWNgðt;cÞ. A 4. Game theory: ‘cave’ psychological world

Our ‘cave’ psychological world is inspired in Platos concrete world, where all things are shadows of the intelligible reality in Plato’s world of thoughts. This world consists of individuals whose taste and crowding types follow the shadows of the idealized taste and crowding types. We propose, for simplicity, that the shadows follow a probability distribution.

We represent byEthe set of all external variableseand we represent byIthe set of all intermediate variablesi. The values of the external and intermediate variables determine the taste typetand the crowding typecof the individuals by a map

t£c:E£I!T £C

ðe;iÞ!ðtðe;iÞ;cðe;iÞÞ:

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The welfare is given by an utility function

ue;i:G£ðN^CÞ^G!R

u_e;iðg;m^GÞ ¼V^gtðe;iÞ;cðe;iÞþf^gtðe;iÞ;cðe;iÞðm^GÞ

where V^gtðe;iÞ;cðe;iÞ measures the satisfaction level that an individual with external and intermediate variables ðe;iÞ has in belonging to a behaviour/group g[G and f^gtðe;iÞ;cðe;iÞðm^GÞ measures the satisfaction level that such an individual has, taking in account, for each crowding typec⁰[C, the number of elementsm^g_c0 that exist in every groupg[G.

We consider a finiteset of shadows K, such that, each shadowk[Kcharacterizes a possible setShðkÞ,E£I of the external and intermediate variables according to a given probability distributionP_k.

We define the shadow representation mapK:S!Kthat associates to each individual s[Sits shadowKðsÞ[K.

The expected utilityEðu_sðg;m^GÞÞof an individuals[Sto choose a behaviour/group g[Gwith a crowding vectorm^Gis given by

Eðusðg;m^GÞÞ ¼EKðsÞhV^gtðe;iÞ;cðe;iÞþEKðSÞf^gtðe;iÞ;cðe;iÞm^Gi

where EKðsÞ is the expected value with respect to the probability distribution PKðsÞ[ ShðKðsÞÞandEKðSÞis the expected value with respect to the joint probability distribution PKðSÞ[ShðKðSÞÞ.

Hence, the expected utility of the ‘cave’ psychological world reflects the effect of the shadows, taking in account their probability distribution.

In this way, when the individuals pass from the intention to the behaviour/group decision, the indetermination of their types is solved arbitrarily, for incidental reasons or reasons that the individuals are not able to predict except in probability. Another possible construction is presented in [2].

The individuals distributionG^* :S!Ginto the different groups is a Bayesian – Nash Equilibrium, if given the choice options of all individuals, no individual feels motivated to change his behaviour/group, i.e. its expected utility does not increase by changing his behaviour/group (see Pinto [12]).

Our new dictionary between our Game Theoretical Model and the Theory of Planned Behaviour, in the ‘cave’ idealized psychological world, is now summarized in Figure 3.

We define theshadow worst neighbours WNgðkÞofk[Kin groupgby

WN_gðkÞ ¼E_k V^gtðe;iÞ;cðe;iÞþ X

c⁰[C;A^g;ctðe;iÞ;cðe;iÞ⁰ ,0

A^g;ctðe;iÞ;cðe;iÞPðc⁰ ⁰ðe;iÞ

2 64

3 75

wherePðc⁰Þis the probability of the realization of the crowding typec⁰.

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We define theshadow best neighboursBNgðkÞofk[K in groupg⁰by

BNgðkÞ ¼Ek V^gtðe;iÞ;cðe;iÞ⁰ þ X

c⁰[C;A^g;ctðe;iÞ;cðe;iÞ⁰ .0

A^g;ctðe;iÞ;cðe;iÞ⁰ Pðc⁰Þ 2

64

3 75:

Lemma5.If WNgðkÞ$BNgðkÞ, for all g⁰[Gng then all the individuals s[K²¹ðkÞwith the same shadow k, choose the same behaviour/group g¼G^*ðSÞ, for every Bayesian – Nash Equilibrium G^* :S!G.

An alternative interesting way to interpretK:S!Kis thatK²¹ðkÞis the number of times that a same individuals[K²¹ðkÞhas to take an action. In this case, one concludes that the individuals[K²¹ðkÞalways choose the same behaviour/group for every Bayesian – Nash equilibrium behaviour/group.

Proof. It follows from the construction of the worst neighboursWNgðkÞand of the best

neighboursBNgðkÞ. A

5. Conclusion

We have described how the theories of planned behaviour or reasoned action study the decision-making mechanisms of individuals and we proposed the Bayesian – Nash equilibria as one, of many, possible mechanisms of transforming human intentions in behaviour. We distinguished two different worlds and described a model associated to Figure 3. Game theoretical model/theory of planned behaviour in the ‘cave’ idealized psychological world.

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each world. We have shown how saturation, boredom and frustration can lead to splitted strategies and how no saturation that can lead to a constant strategy.

Acknowledgements

We thank the Programs POCTI and POSI by FCT and Ministe´rio da Cieˆncia, Tecnologia e do Ensino Superior, Calouste Gulbenkian Foundation and Centro de Matema´tica da Universidade do Minho (CMAT) for their financial support.

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