DYNAMICS OF HUMAN DECISIONS

We study a dichotomous decision model, where individuals can make the decision yes or no and can influence the decisions of others. We characterize all decisions that form Nash equilibria. Taking into account the way individuals influence the decisions of others, we construct the decision tilings where the axes reflect the personal preferences of the individuals for making the decision yes or no. These tilings characterize geometrically all the pure and mixed Nash equilibria. We show, in these tilings, that Nash equilibria form degenerated hystereses with respect to the replicator dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hystereses can help to explain the sudden appearance of social, political and economic crises. We observe the existence of limit cycles for the replicator dynamics associated to situations where the individuals keep changing their decisions along time, but exhibiting a periodic repetition in their decisions. We introduce the notion of altruist and individualist leaders and study the way that the leader can affect the individuals to make the decision that the leader pretends.


1.
Introduction. The main goal in Planned Behavior or Reasoned Action theories, as developed in the works of Ajzen (see [1]) and Baker (see [4]), is to understand and predict the way individuals turn intentions into behaviors. Almeida-Cruz-Ferreira-Pinto (see [2,9]) developed a game theoretical model for reasoned action, inspired by the works of J. Cownley and M. Wooders (see [7]). Here, we study the Pinto's dichotomous decision model (see [9,11]), which is a simplified version of the Almeida-Cruz-Ferreira-Pinto decision model. In this model, there are just two types t ∈ {t 1 , t 2 } of individuals and two possible decisions d that individuals can make. In this case, they have to choose between yes or no, i.e. d ∈ {Y, N }. The yes-no decision model incorporates, in the coordinates of the preference decision matrix how much an individual with type t 1 or with type t 2 likes or dislikes, to make a decision d ∈ {Y, N }. The yes-no decision model incorporates, in the coordinates of the preference neighbors and no-neighbors matrices, the preference that individuals with a certain type t i have for other individuals, with the same or a different type t j , to make the same decision or the opposite decision as theirs (see [5,10]). The preference decision matrix and the neighbors and no-neighbors matrices can be very complex to find explicitly in real cases because they encode, for instance, information from economic, educational, political, psychological and social variables. However, if we have a qualitative or rough knowledge of these matrices, we can obtain relevant information on how individuals make decisions and why to make decisions can be so complex.
We characterize all the pure Nash equilibria and we show that the pure Nash equilibria are, in general, asymptotically stable with respect to the replicator dynamics. The pure Nash equilibria are either cohesive, i.e. all individuals with the same preferences make the same decision, or disparate, i.e. there are individuals with the same preferences that make opposite decisions. The disparate pure Nash equilibria can correspond to conflicting decisions that divide a community. We characterize all the strict mixed Nash equilibria and we prove that the strict mixed Nash equilibria are, in general, unstable. Fixing the parameters of the preference neighbors matrices we construct tilings in the plane, where the horizontal axis represent the relative preference of individuals with type t 1 to make the decision yes or no, the vertical axis represent the relative preference of individuals with type t 2 to make the decision yes or no and the pure and mixed Nash equilibria form the tiles. We prove that the tilings give a full geometrically characterization of the pure and mixed Nash equilibria.
We say that individuals with a certain type t j have a positive influence over individuals with the same or other type t i if the individuals with type t i prefer to make the same decision as the individuals with type t j and we say that individuals with type t j have a negative influence over individuals with type t i if the individuals with type t i prefer to make the opposite decision from the individuals with type t j . If all the individuals have a positive influence over individuals with the same type then there are no disparate Nash equilibria. However, if there are individuals with a certain type t j that have a negative influence over individuals with the same type t j then there are disparate Nash equilibria that are asymptotically stable.
The stable manifolds of the strict mixed Nash equilibria can be locally characterized by appropriate symmetries of the model. They are the main reason for certain decision strategies to persist for long periods of time before breaking down and converge to quite different strategies of decision. They also explain, partially, the complexity of the non-intuitive successive reversal of the individuals decisions along time before converging to a stable equilibrium, i.e. a high number of individuals have to keep modifying their decisions through the transient dynamics before reaching the equilibrium. Furthermore, we observe the existence of stable periodic orbits for the replicator dynamics, i.e. the individuals decisions keep changing along time exhibiting a periodic pattern. The replicator dynamics equilibria form hystereses that provide insight into how small changes in economic, educational, political, DYNAMICS OF HUMAN DECISIONS 3 psychological or social variables can reverse abruptly individual and collective decisions. These changes in the collective decisions can lead to serious political, social or economic transformations in society.
Following [3], we introduce the notion of altruist and individualist leaders. The altruist and individualist leaders offer, respectively, advantages or disadvantages to the individuals of both type. The leader is biased if the leader offers advantages to individuals with a certain type and disadvantages to individuals with the other type. The individuals can increase or decrease the advantages or disadvantages offered by the leader depending upon their own abilities characterized by their type. We study the way that the leader can affect the individuals (potential followers) to make the decision that the leader pretends.
2. Yes-No Decision Model. As in [11], the yes-no decision model has two types T = {t 1 , t 2 } of individuals. Let I 1 = {1, . . . , n 1 } be the set of all individuals with type t 1 , and let I 2 = {1, . . . , n 2 } be the set of all individuals with type t 2 . Let I = I 1 I 2 . The individual i ∈ I has to make one decision d ∈ D = {Y, N } 1 . Let L be the preference decision matrix whose coordinates γ d p indicate how much an individual with type t p likes or dislikes, to make decision d The preference decision matrix indicates for each type of individuals the decision that the individuals prefer, i.e. the taste type of the individuals (see [2,6,7,9,11]). Let N d be the preference neighbors matrix whose coordinates β d pq indicate how much an individual with type t p who decides d likes or dislikes that an individual with type t q also makes decision d Let N d be the preference non-neighbors matrix whose coordinates β d pq indicate how much an individual with type t p who decide d, likes or dislikes that an individual with type t q makes decision d = d The preference neighbors and non-neighbors matrices indicate, for each type of individuals whose decision is d, whom they prefer, or do not prefer, to be with in each decision, i.e. the crowding type of the individuals (see [2,6,7,9]). We describe the (pure) decision of the individuals by a (pure) strategy map S : I → D that associates to each individual i ∈ I its decision S(i) ∈ D. Let S be the space of all strategies S. Given a strategy S, let O S be the strategic decision matrix whose coordinates l d p = l d p (S) indicate the number of individuals with type t p , who make decision d The strategic decision vector associated to a strategy S is the vector (l 1 , l 2 ) = (l y 1 (S), l y 2 (S)). Hence, l 1 (resp. n 1 − l 1 ) is the number of individuals with type t 1 who make the decision Y (resp. N ). Similarly, l 2 (resp. n 2 − l 2 ) is the number of individuals with type t 2 who make the decision Y (resp. N ). The set O of all possible strategic decision vectors is Let: Let U 1 : D × O → R the utility function of an individual with type t 1 be given by Let U 2 : D × O → R the utility function of an individual with type t 2 be given by Given a strategy S ∈ S, the utility U i (S) of an individual i with type t p(i) is given by U p(i) (S(i); l y 1 (S), l y 2 (S)).
be the horizontal relative decision preference of the individuals with type t 1 and let y = ω Y 2 − ω N 2 be the vertical relative decision preference of the individuals with type t 2 . Let A ij = α Y ij + α N ij , for i, j ∈ {1, 2}, be the coordinates of the influence matrix.
If x > 0, the individuals with type t 1 prefer to decide Y , without taking into account the influence of the others. If x = 0, the individuals with type t 1 are indifferent to decide Y or N , without taking into account the influence of the others. If x < 0, the individuals with type t 1 prefer to decide N , without taking into account the influence of the others.
If A ij > 0, the individuals with type t j have a positive influence over the utility of the individuals with type t i . If A ij = 0, the individuals with type t j are indifferent for the utility of the individuals with type t i . If A ij < 0, the individuals with type t j have a negative influence over the utility of the individuals with type t i .
3. Cohesive Nash equilibria. We will show that the relative decision preferences and the influence matrix together with the total number of individuals of each type encode all the relevant information for characterizing the Nash equilibria.
A strategy S * : I → D is a (pure) Nash equilibrium if for every individual i ∈ I and for every strategy S ∈ S, with the property that S * (j) = S(j) for every individual j ∈ I \ {i}. The Nash domain N(S) of a strategy S ∈ S is the set of all pairs (x, y) for which S is a Nash equilibrium.
Definition 3.1. A cohesive strategy 2 is a strategy in which all individuals with the same type prefer to make the same decision. A disparate strategy is a pure strategy that is not cohesive.
As in [11], we construct the Nash domains N(S) for the cohesive strategies. We observe that there are four cohesive strategies: (Y, Y ) strategy: all individuals make the decision Y ; (Y, N ) strategy: all individuals, with type t 1 , make the decision Y , and all individuals, with type t 2 , make the decision N ; (N, Y ) strategy: all individuals, with type t 1 , make the decision N and all individuals, with type t 2 , make the decision Y ; (N, N ) strategy: all individuals make the decision N . The The horizontal H(Y, N ) and vertical V (Y, N ) strategic thresholds of the (Y, N ) strategy are

4.
Disparate Nash equilibria. An (l 1 , l 2 ) strategic set is the set of all pure strategies S ∈ S with l 1 (S) = l 1 and l 2 (S) = l 2 . An (l 1 , l 2 ) cohesive strategic set is an (l 1 , l 2 ) strategy set with l 1 ∈ {0, n 1 } and l 2 ∈ {0, n 2 }. An (l 1 , l 2 ) disparate strategic set is an (l 1 , l 2 ) strategic set that is not cohesive. We observe that a cohesive strategic set has a single strategy and a disparate strategic set has more than one strategy.
Since individuals with the same type are identical, a strategy to be a Nash equilibrium depends only upon the number of individuals of each type that decide either Y or N , and not upon the individual who is making the decision.
Definition 4.1. An (l 1 , l 2 ) pure Nash equilibrium (set) is an (l 1 , l 2 ) strategic set whose strategies are Nash equilibria. The (pure) Nash domain N(l 1 , l 2 ) is the set of all pairs (x, y) for which the (l 1 , l 2 ) strategic set is a Nash equilibrium set.
Hence, if the individuals with a given type have a positive influence over the utility of the individuals with the same type, i.e. A 11 > 0 and A 22 > 0, then there are no disparate Nash equilibria.
Proof. Suppose, by contradiction, that the (l 1 , l 2 ) strategy is a Nash equilibrium for l 1 ∈ {1, . . . , n 1 − 1}. Hence, the following two inequalities hold By rearranging the terms in the previous inequalities, we obtain A 11 ≤ 0 which contradicts that A 11 is positive. Hence, Lemma 4.2 (i) holds. The proof of the other cases follow similarly to the proof of the first case.
The cohesive horizontal vector H is and the disparate Nash domain N(l 1 , n 2 ) is given by and the disparate Nash domain N(n 1 , l 2 ) is given by Hence, if the individuals with a given type have a non-positive influence over the utility of the individuals with the same type, i.e. A 11 ≤ 0 and A 22 ≤ 0, then for every (l 1 , l 2 ) disparate strategic set there are relative preferences for which (l 1 , l 2 ) is a Nash equilibrium set.
Proof. The (l 1 , 0) strategy is a Nash equilibrium if, and only if, the following three inequalities hold . Hence, the proof of Lemma 4.3 (i) follows from rearranging the terms in the previous inequalities. The proof of Lemma 4.3 (ii) follows similarly to the proof of the first case. Let us prove Lemma 4.3 (iii). The (l 1 , l 2 ) strategy is a Nash equilibrium if, and only if, the following four inequalities hold . Hence, the proof of Lemma 4.3 (iii) follows from rearranging the terms in the previous inequalities.

5.
Mixed Nash equilibria. Recall that I = I 1 I 2 . We describe the (mixed) decision of the individuals by a (mixed) strategy map S : I → [0, 1] that associates to each individual i ∈ I 1 the probability p i = S(i) to decide Y ∈ D and to each individual j ∈ I 2 the probability q j = S(j) to decide Y ∈ D. Hence, each individual i ∈ I 1 decides N ∈ D with probability 1 − p i = 1 − S(i) and each individual j ∈ I 2 decides N ∈ D with probability 1 − q j = 1 − S(j). We assume that the decisions of the individuals are independent. Define For every individual j ∈ I 2 , the utility function Proof. The proof follows by induction on the number of individuals. Let (n 1 , n 2 ) = (1, 1). The utility function of individual i = 1, with type t 1 , is given by By substituting the fitness functions in the previous identity, we obtain After rearranging the terms, the last identity becomes Similarly, the utility function of individual j = 1, with type t 2 , is given by Let us add one more individual i = n 1 + 1, with type t 1 , and compute its utility function. Let us suppose, by induction, that the utility functions are known for n 1 individuals with type t 1 and for n 2 individuals with type t 2 . Let P = n1 k=1 p k , Q = n2 k=1 q k and P = P + p n1+1 . The utility function of the individual n 1 + 1 is given by Thus, . By substituting the fitness functions in the previous identity, we obtain Hence, The proof follows similarly if we add one more individual j = n 2 + 1 with type t 2 and compute its utility function.
A strategy S * : for every individual i ∈ I and for every strategy S ∈ S with the property that S * (j) = S(j), for every individual j ∈ I \ {i}.
Hence, if A 11 = 0, then there is not a mixed Nash equilibrium with the property that 0 < p i1 = p i2 < 1. Furthermore, if A 22 = 0, then there is not a mixed Nash equilibrium with the property that 0 < q j1 = q j2 < 1.
which implies Lemma 5.2 (i). The proof of Lemma 5.2 (ii) follows similarly.
Since individuals with the same type are identical, if a mixed strategy contained in the (l 1 , k 1 , p; l 2 , k 2 , q) mix strategic set is a Nash equilibrium, then all the strategies in the (l 1 , k 1 , p; l 2 , k 2 , q) mixed strategic set are Nash equilibria. Definition 5.3. An (l 1 , k 1 , p; l 2 , k 2 , q) mixed Nash equilibrium (set) is an (l 1 , k 1 , p; l 2 , k 2 , q) strategic set whose strategies are Nash equilibria. The (mixed) Nash domain N(l 1 , k 1 , p; l 2 , k 2 , q) is the set of all pairs (x, y) for which the (l 1 , k 1 , p; l 2 , k 2 , q) strategic set is a mixed Nash equilibrium set.
Let J 0 = (−∞, 0) and J n1 = J n2 = [0, +∞). (iii): For p, q ∈ (0, 1), the mixed Nash domain N(0, n 1 , p; 0, n 2 , q) is the singleton By Theorem 5.4, if l 1 = n 1 , then the only strict mixed Nash equilibria are the ones presented in (i). Furthermore, if l 1 = 0, then the only strict mixed Nash equilibria are the ones presented in (ii) and (iii). Hence, if the individuals with a given type have a positive influence over the utility of the individuals with the same type, i.e. A 11 > 0 and A 22 > 0, then there are no mixed Nash equilibrium, unless all the individuals with the same type opt for a mixed strategy.
In Figure 1, we show the geometric interpretation of Theorem 5.4, with − → Proof. The mixed strategy (0, 0, 0; 0, n 2 , q) is a Nash equilibrium if, and only if, the following two inequalities hold We note that U 2 (q; P, Q) ≥ U 2 (q ; P, Q − q + q ) , if, and only if, f Y,2 (q; P, Q) = f N,2 (q; P, Q) .

DYNAMICS OF HUMAN DECISIONS
We note that Hence, by Lemma 5.2, Thus, the proof of Theorem 5.4 (i) follows from rearranging the terms in the previous inequalities. The proof of Theorem 5.4 (ii) follows similarly to the proof of Theorem 5.4 (i). Let us prove Theorem 5.4 (iii). The mixed strategy (0, n 1 , p; 0, n 2 , q) is a Nash equilibrium if, and only if, the following two inequalities hold U 1 (p; P, Q) ≥ U 1 (p ; P − p + p , Q) , U 2 (q; P, Q) ≥ U 2 (q ; P, Q − q + q ) .
Thus, by Lemma 5.2, Hence, Theorem 5.4 (iii) follows from rearranging the terms in the previous inequalities.
By Theorem 5.5, if l 1 = n 1 , then the only strict mixed Nash equilibria are the ones presented in (i). Furthermore, if l 1 = 0, then the only strict mixed Nash equilibria are the ones presented in (ii) and (iii). Hence, if the individuals with type t 1 have a positive influence over the utility of the individuals with the same type, i.e. A 11 > 0, then, for every Nash equilibrium all the individuals with the type t 1 opt for the same strategy either pure or mixed.
Lemma 5.7. Let S be a strategy given by S(i) = p i , for i ∈ I 1 , and S(j) = q j , for j ∈ I 2 .
Proof. It follows from putting together Lemma 5.2 with Theorems 5.5 and 5.6.
In the following remark, we observe, for a mixed Nash equilibria, who receives higher utility between (i) the individuals who decide Y, (ii) the individuals who decide N, and (iii) the individuals who decide based on probability.
6. Replicator Dynamics. Recall that I = I 1 I 2 and that a strategy S : I → [0, 1] associates to each individual i ∈ I 1 the probability p i = S(i) to decide Y ∈ D and to each individual j ∈ I 2 the probability q j = S(j) to decide Y ∈ D. Recall that Hence, the replicator dynamicsṠ = G(S; x, y) can be rewritten as We observe that (i) if p i (0) < p j (0), then p i (t) < p j (t); (ii) if p i (0) = p j (0), then p i (t) = p j (t); (iii) if q i (0) < q j (0), then q i (t) < q j (t); (iv) if q i (0) = q j (0), then q i (t) = q j (t); for every t ∈ R.
A strategy S : The coefficients of the linearized replicator dynamics DG(S; x, y) are An equilibrium strategy S is (strongly) stable if all the eigenvalues of the linearized replicator dynamics DG(S; x, y) have real parts that are negative. An equilibrium strategy S is (strongly) unstable if there is at least one eigenvalue of the linearized replicator dynamics DG(S; x, y) with positive real part.
Hence, if A 11 = 0, then there is not a dynamical equilibrium with the property that 0 < p i1 = p i2 < 1. Furthermore, if A 22 = 0, then there is not a dynamical equilibrium with the property that 0 < q j1 = q j2 < 1.
Proof. The proof is analogous to the proof of Lemma 5.2.
Hence, assuming that A 11 = 0 and A 22 = 0, the dynamical equilibria of the replicator dynamics are contained in the union of all (l 1 , k 1 , p; l 2 , k 2 , q) strategic sets, where p, q ∈ [0, 1], 0 ≤ l 1 + k 1 ≤ n 1 and 0 ≤ l 2 + k 2 ≤ n 2 . Thus, to find and study the dynamical equilibria we introduce the following notation: • Let us define V [1] and V [2] as follows: k=1 v 2r k . In this notation, the replicator dynamics DG(v; x, y) are given by the following n 1 + n 2 ODE: where m 1 , m 2 ∈ {l, m, r} and (x, y) ∈ R 2 . Definition 6.2. The (l 1 , k 1 , p; l 2 , k 2 , q) canonical strategy is defined as follows: • for all i ∈ {1, . . . , l 1 }, j ∈ {1, . . . , n 1 − (l 1 + k 1 )} and k ∈ {1, . . . , The linearized replicator dynamics DG = DG(l 1 , k 1 , p; l 2 , k 2 , q; x, y) at the (l 1 , k 1 , p; l 2 , k 2 , q) canonical strategy is given by the matrix where the coefficients are matrices with the following coordinates: Definition 6.3. The equilibria domain E(l 1 , k 1 , p; l 2 , k 2 , q) is the set of all pairs (x, y) ∈ R 2 for which the strategies contained in the (l 1 , k 1 , p; l 2 , k 2 , q) strategic set are equilibria of the replicator dynamics. The (strongly) stable domain S(l 1 , k 1 , p; l 2 , k 2 , q) is the set of all pairs (x, y) ∈ R 2 for which the strategies contained in the (l 1 , k 1 , p; l 2 , k 2 , q) strategic set are (strongly) stable equilibria of the replicator dynamics. The (strongly) unstable domain U(l 1 , k 1 , p; l 2 , k 2 , q) is the set of all pairs (x, y) ∈ R 2 for which the strategies contained in the (l 1 , k 1 , p; l 2 , k 2 , q) strategic set are (strongly) unstable equilibria of the replicator dynamics.
We observe that the stable domains are contained in the Nash domains and the Nash domains are contained in the equilibria domains, i.e. S(l 1 , k 1 , p; l 2 , k 2 , q) ⊆ N(l 1 , k 1 , p; l 2 , k 2 , q) ⊆ E(l 1 , k 1 , p; l 2 , k 2 , q) .
For the (l 1 , l 2 ) (pure) strategic set, the dynamic equilibria set coincides with R 2 , i.e. E(l 1 , l 2 ) = R 2 . The following geometric relation associates to every pair (x, y) ∈ R 2 a unique pair (p, q) ∈ R 2 and vice-versa: Theorem 6.4. The eigenvalues of DG(l 1 , l 2 ; x, y) = DG(l 1 , 0, 0; l 2 , 0, 0; x, y) are The eigenspaces of DG(l 1 , l 2 ; x, y) are Furthermore, all the other coefficients of DG are equal to 0. Hence, DG is a diagonal matrix. The eigenvalues and the eigenspaces of DG are the ones presented in (2) and (3). Hence, the proof of the second part of Theorem 6.4 follows from applying Lemma 4.3.

Proof
Furthermore, Hence, the space spaned by w 1 and w 2 is DG invariant and DG| < w 1 , w 2 > is given by the matrix M (k 1 , k 2 ). Therefore, the eigenvalues of DG| < w 1 , w 2 > are the eigenvalues of the matrix M (k 1 , k 2 ). For t ∈ {1, 2} and s ∈ {l, m, r}, the vectors e i [ts] − e j [ts], with i = j, belong to the eigenspace of the eigenvalue λ[ts]. Since the space spaned by w 1 and w 2 is DG invariant, the vectors e i [ts] belong to the extended eigenspace of the eigenvalue λ[ts].
In Figure 4, we observe the appearance of periodic cycles for the replicator dynamics. The individuals keep modifying their decisions along time exhibiting a periodic pattern in their decisions. 7. Leadership Model. A leader is an individual who can influence others to make a certain decision. For simplicity, we assume that the leader will influence other individuals to make decision Y. We study how the choice of the leader can influence the potential followers (individuals of type t 1 and t 2 ) to make the decision he pretends, see [3].
As in [3], the parameters (θ i , P i , L i ), with i ∈ {1, 2}, characterize the leaders and the potential followers. The types of leaders are defined as follows: • Altruistic, individualist and biased leaders. The leader donates P 1 to the individuals of type t 1 and P 2 to the individuals of type t 2 . The altruistic leader, for the individuals with type t i , is the one who distributes a valuation to potential followers making the decision Y, i.e. P i > 0; while the individualist leader, for the individuals with type t i , is the one who distributes a devaluation or debt to potential followers making the decision Y, i.e. P i < 0. The biased leader is the one who distributes a valuation to one type of potential followers and a debt to the other type of potential followers, i.e. P 1 P 2 < 0.
• Consumption or wealth creation by the followers. Define θ 1 , θ 2 as the parameters of the consumption or wealth creation on the valuation distributed by the leader to other individuals. Therefore, the new valuation of the individuals, with type t 1 and t 2 , to make decision Y is given by where ω Y i corresponds to the valuation before the influence of the leader of the individuals to make decision Y. There is wealth creation by the followers of type t i when P i > 0 and θ i > 1 or when P i < 0 and θ i < 1. There is wealth consumption by the followers of type t i when P i > 0 and θ i < 1 or when P i < 0 and θ i > 1.
• Influential and persuasive leader. The influence or persuasiveness of the leader on other individuals is measured by the parameters L 1 and L 2 . The individuals have a new valuation, when they make the decision N, under the influence of the leader, given by ω N i − L i . If L i < 0, the individuals with type t i will like more to make the decision that the leader pretends; however, if L i > 0, the individuals with type t i will like more to make the opposite decision from the one that the leader pretends. (ii) If θ2P2 n2 + L 2 > V − y, then the individuals with type t 2 make the decision Y.
As in [3], the inequalities above provide a sufficient condition in the values of the donated parts P 1 and P 2 , in the values of the influence and persuasiveness L 1 and L 2 of the leader and in the values of the consumption or creation of wealth θ 1 and θ 2 by the followers, implying that the potential followers make the same decision as the leader.
8. Conclusions. The union of the equilibria E(l 1 , k 1 , p; l 2 , k 2 , q) form hystereses. The equilibria S(l 1 ; l 2 ) are the stable part of the hystereses and the equilibria U (l 1 , k 1 , p; l 2 , k 2 , q) are the unstable part of the hystereses. The equilibria in E(l 1 , k 1 , p; l 2 , k 2 , q)\ {S(l 1 , k 1 , p; l 2 , k 2 , q), U (l 1 , k 1 , p; l 2 , k 2 , q)} can be stable or unstable equilibria. Hence, small changes in the coordinates of the influence matrix that determines the equilibria sets S(l 1 ; l 2 ) and U (l 1 , k 1 , p; l 2 , k 2 , q) can create or annihilate cohesive and 24 RENATO SOEIRO, ABDELRAHIM MOUSA, TÂNIA OLIVEIRA AND ALBERTO PINTO disparate Nash equilibria giving rise to abrupt and collective changes in the decisions of the individuals that are explained by the hystereses. Furthermore, we observed the appearance of periodic attracting cycles and so the individuals can keep changing their decisions with a periodic pattern. We demonstrated how the characteristics of the leader can have a positive or negative influence over the decisions of the individuals. In particular, we show that an individualist leader might have to be more persuasive than an altruistic leader to convince the individuals to make a particular decision.