GMCR with stochastic preferences involves exactly the same components of the described GMCR, with the exception that DMs now are considered to have stochastic preference over states. Thus, it is assumed that every DM in a conflict is able to indicate the probability P (a, b) that he prefers state a over state b, for every pair of states a and b in a conflict. The intuition of this idea is that individuals can use empirical data to establish a probability distribution over the states. Or, in some conflict situations, decision makers, who may be individuals, but also institutions or countries, may have more than one representant and each representant has the possibility of being chosen the actual DM representant in a particular conflict with a certain probability. Whereas each representant may have a standard deterministic preference relation over the set of states, before the actual representant is chosen we cannot express the DM’s preference by a standard preference relation, but we could use an stochastic preference relation defining that the probability that the DM prefers state a over b as the sum of the probabilities of the representants that prefer a over b.
Consider the following example that illustrate this situation.
Example 3.1.
We present a modified version of a hypothetical conflict proposed by Hipel (2001) that illustrates the usefulness of the proposed GMCR with stochastic preferences. In this conflict there are two DMs:
Environmentalists E and Developers D. Environmentalists can choose to be proactive P in promoting environmental responsibility or not, in which case they are called reactive.
R. Developers can choose between being sustainable S or not, which is represented by U . Since there are only two options, there are 22 = 4 possible combinations: (P, S), (P, U ), (R, U ) and (R, S), which are the possible states of the conflict. If DM E, for example, is in state (P, S), he can either stay in (P, S) or change to (R, S), while DM D, in the same state, can stay in (P, S) or change to (P, U ). The conflict is modeled as having an initial state and evolves depending on whether the DMs change actions and therefore states.
The original example assumes that preferences are deterministic and E’s preferences are such that (P, S) >-E (R, S) >-E (P, U ) >-E (R, U ) and considers that there are two types of DM D: one
gives low priority to environmentalism and the other is more responsible in this sense than the first one.
The preferences of the first type, DU , are (R, U ) >-DU (P, U ) >-DU (R, S) >-DU (P, S) and second, DS, are such that (R, S) >-DS (P, S) >-DS (R, U ) >-DS (P, U ).
We modify the original example by considering that if a DM i deterministically prefers one state sp to another sq , then we say that Pi(sp, sq ) = 1. Otherwise, we say that the probability is zero. According to this definition, the likelihood that E prefers one state to another are shown in Table 1. Each cell expresses the probability that DM E prefers the line state to the column state.
Table 1: Stochastic Preferences for DM E E (P,S) (R,S) (P,U) (R,U) (P,S) 0.00 1.00 1.00 1.00 (R,S) 0.00 0.00 1.00 1.00 (P,U) 0.00 0.00 0.00 1.00 (R,U) 0.00 0.00 0.00 0.00
Consider now that DM j can be of several different types and each type has a possibly different preference order. Let us also assume that there is a probability distribution over the set of types that represent DM j in the conflict. In this case, we say that DM j has a stochastic preference for state sp over state sq that is given by the sum of the probabilities of the types of DM j that prefer sp over sq . Thus, for example, consider the probability distribution in which P (D = DS ) = 0.30 and P (D = DU ) = 0.70. Thus, the stochastic preferences of DM D are as shown in Table 2, where each cell indicates the likelihood that DM D prefers the line state to the column state. For example, PD ((R, S), (P, S)) = 0.30 + 0.70 = 1, since the two possible types of DM D prefer (R, S) over (P, S), while PD ((P, U ), (P, S)) = 0.70, because the likelihood of the type of DM D that prefers (P, U ) over (P, S) is 0.70 , i.e., the probability of the type being DU is 0.70.
Table 2: Stochastic Preferences for DM D D (P,S) (R,S) (P,U) (R,U) (P,S) 0.00 0.00 0.30 0.30 (R,S) 1.00 0.00 0.30 0.30 (P,U) 0.70 0.70 0.00 0.00 (R,U) 0.70 0.70 1.00 0.00
GMCR model contains a set of DMs, N . Each DM has a set of actions available. From this set, each DM must choose one action and this choice affects the current state of the conflict. The set of states that DM i can achieve when the current state of the conflict is s is denoted by Ri(s).
Let us consider now that the DMs participating in the conflict prefer a state over the other
according to a probability distribution, so that given any two states sp and sq , any DM i has a probability that expresses the chance he prefers state sp over sq which is denoted by Pi(sp, sq ).
Thus, considering the new preference structure that allows each DM to have stochastic prefer- ence over states, we can define new concepts of stability that makes use of this extra information revealed by the preference structure. We propose notions of stability which are based on the standard notions of Nash stability, Metarationality and Sequential Stability.
Notions of Stability in GMCR with Stochastic Preferences In this section, consider parameters α, β and γ lying in the interval [0, 1].
α-Nash Stability. In the original GMCR model, a state being Nash stable means that none of the DMs could move to a state that is better than the equilibrium one. In the present model, we want to have a stability notion that captures the intuition that no DM can move to a state which is preferred over the equilibrium state with a sufficiently high probability. The following definition formalizes this idea:
Definition 3.2.: Let i ∈ N . A state s ∈ S is α−Nash stable for DM i, if for every state s1∈ Ri(s), Pi(s1, s) ≤ 1 − α.
Thus, a state s is α-Nash stable for DM i if among all the states that i can achieve when he is in s there is no state that he prefers to s with probability greater than 1 − α. For example, a state s that is 0.9-Nash stable for DM i is such that among all the states that i can achieve from s there is none that i prefers to s with probability greater than 0.10. As another example, a state is 1.0-Nash stable for DM i if among all the states that i can achieve from s there is none he prefers over s with probability greater than zero.
As we show in the next section, the set of α-Nash stable states monotonically decreases as one increases α.
Thus, for each state there exists an interval of values of α for which such state satisfies α-Nash stability. The supremum of such interval characterizes the intensity of Nash stability of the corresponding state. In order to guarantee that each state can be associated with such a number, there must be at least one α such that the state satisfies α-Nash stability. This is the reason why we allow α to be equal to zero, since every state is obviously 0-Nash stable.
(α, β)-Metarationality. In the original GMCR model, a state being metarational stable means that even if one DM could move to a more preferred state there would be a counter move from another DM that would lead the conflict to a state worse than the equilibrium one for the first DM. In the present model, we want to have a stability notion that captures the intuition that even if one DM can move to a state which is preferred over the equilibrium state with a sufficiently high probability, there would be a counter move from another DM leading to a state which is not much more preferred than the equilibrium one by the first DM.
The following definition formalizes this idea:
Definition 3.3.: Let i ∈ N . A state s ∈ S is (α, β)-Metarational for DM i if for all s1 ∈ Ri(s) such that Pi(s1, s) > 1 − α, there exists s2 ∈ Rj (s1) such that Pi(s2, s) ≤ 1 − β.
Thus, a state s is (α, β)-Metarational stable for DM i if for every state that i can achieve from s and he prefers to s with probability greater than 1−α, then there is a response moving to a state s2 that some other DM j can make such that i does not prefer s2 over s with probability greater than 1 − β. For example, a state s that is (0.9, 0.7)-Metarational stable for DM i is such that for every state that i can achieve from s and that he prefers over s with probability greater than 0.10, there is a response moving to a state s2 that some other DM j can make such that i does not prefer s2 over s with probability greater than 0.3. As another example, a state is (1, 1)-Metarational stable for DM i if for every state that i can achieve from s and that he prefers over s with positive probability, there is a response moving to a state s2 that some other DM j can make such that i does not prefer s2 over s with positive probability.
As we show in the next section, the set of (α, β)-Metarational stable states monotonically decreases as one increases either α or β. Thus, for each state there exists a region of (α, β) values for which such state satisfies (α, β)-Metarational stability. Such region characterizes the intensity of Metarational stability of the corresponding state. In some contexts, it may make sense to add a restriction that α = β, but since we do not see any compelling reason to do so in general we add such parameter to give more flexibility. In this case, α defines the threshold for the preference when we are analyzing moves of a DM that is considering moving away from an equilibrium state and β defines the threshold for the preference when we analyze the counter move of a different DM.
(α, β)-Symmetric Metarationality. In the original GMCR model, a state being symmetric metarational stable means that even if one DM could move to a more preferred state there would be a counter move from another DM that would lead the conflict to a state worse than the equilibrium one for the first DM and, moreover, the first DM can not move away from such latter state to another one that is preferred to the equilibrium state. In the present model, we want to have a stability notion that captures the intuition that even if one DM can move to a state which is preferred over the equilibrium state with a sufficiently high probability, there would be a counter move from another DM leading to a state which is not much more preferred than the equilibrium one by the first DM and, moreover, the first DM can not move away from such latter state to another one that is preferred over the equilibrium state with sufficiently high probability.
The following definition formalizes this idea:
Definition 3.4.: Let i ∈ N . A state s ∈ S is (α, β)-Symmetric Metarational for DM i if for all s1 ∈ Ri(s) such that Pi(s1, s) > 1 − α, there exists s2 ∈ Rj (s1) such that Pi(s2, s) ≤ 1 − β and there is no s3 ∈ Ri(s2) such that Pi(s3, s) > 1 − α.
Thus, a state s is (α, β)-Symmetric Metarational stable for DM i if for every state that i can achieve from s and he prefers to s with probability greater than 1 − α, then there is a response moving to a state s2 that some other DM j can make such that i does not prefer s2 over s with probability greater than 1 − β and, moreover, there is no counter response that i can make from s2 leading to a state s3 such that i prefers s3
over s with probability greater than 1 − α.
For example, a state s that is (0.9, 0.7)-Symmetric Metarational stable for DM i is such that for every state that i can achieve from s and that he prefers over s with probability greater than 0.10, there is a response moving to a state s2 that some other DM j can make such that i does not prefer s2 over s with probability greater than 0.30 and, moreover, there is no counter response that i can make from s2 leading to a state s3
such that i prefers s3 over s with probability greater than 0.10. As another example, a state is (1, 1)-Symmetric Metarational stable for DM i if for every state that i can achieve from s that he prefers over s with positive probability, there is a response moving to a state s2 that some other DM j can make such that i does not prefer s2 over s with positive probability and, moreover, there is no counter response that i can make from s2 leading to a state s3 such that i prefers s3 over s with positive probability.
As we show in the next section, the set of (α, β)-Symmetric Metarational stable states monotonically decreases as one increases either α or β. Thus, as in the case of metarationality, for each state there exists a region of (α, β) values for which such state satisfies (α, β)-Symmetric Metara- tional stability. Such region characterizes the intensity of Symmetric Metarational stability of the corresponding state. We also do not see any compelling reason to require that in general α = β and such parameters have exactly the same interpretation as in the previous case.
(α, β, γ)-Sequential Stability. In the original GMCR model, a state being sequentially stable means that even if one DM could move to a more preferred state there would be a counter move from another DM that would lead the conflict to a state worse than the equilibrium one for the first DM and, moreover, such counter move is not harming to the second DM. In the present model, we want to have a stability notion that captures the intuition that even if one DM can move to a state which is preferred over the equilibrium state with a sufficiently high probability, there would be a counter move from another DM leading to a state which is not much more preferred than the equilibrium one by the first DM and, moreover, such counter move of the second DM leads him to a state that is preferred over the current state with a sufficiently high probability. The following definition formalizes this idea:
Definition 3.5.: Let i ∈ N . A state s ∈ S is (α, β, γ)-Sequentially Stable for DM i if for all s1 ∈ Ri(s) such that Pi(s1, s) > 1 − α, there exists s2 ∈ Rj (s1) such that Pj (s2, s1) ≥ γ and Pi(s2, s) ≤ 1 − β.
Thus, a state s is (α, β, γ)-Sequentially Stable for DM i if for every state that i can achieve from s and he prefers with probability greater than 1 − α, then there is a response moving to a state s2 that some other DM j can make such that j prefers s2 over s1 with probability greater than or equal to γ and i does not prefer s2 over s with probability greater than 1 − β.
For example, a state s that is (0.9, 0.8, 0.7)-Sequentially Stable for DM i is such that for every state s1 that I can achieve from s and that he prefers over s with probability greater than 0.1, there is a response moving to a state s2 that some other DM j can make such that j prefers s2 over s1 with probability greater than or equal to 0.7 and i does not prefer s2 over s with probability greater than 0.2. As another example, a state is (1, 1, 1)-Sequentially Stable for DM i if for every state s1 that i can achieve from s that he prefers with positive probability, there is a response moving to a state s2 that some other DM j can make such that j prefers s2 over s1 with probability one and i does not prefer s2 over s with positive probability.
As we show in the next section, the set of (α, β, γ)-Sequentially stable states monotonically decreases as one increases either α, β or γ. Thus, for each state there exists a region of (α, β, γ) values for which such state satisfies (α, β, γ)-Sequential stability. Such region characterizes the intensity of Symmetric Metarational stability of the corresponding state. We also do not see any compelling reason to add any correlation for the values of the parameters α, β and γ. α and β have exactly the same interpretation as in the previous cases while γ is interpreted as the threshold for the preference when we are analyzing moves of the second DM that is considering to make a counter move.