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Nash Equilibrium

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Nash equilibrium is a situation in which all subjects are trying to maximize their own profits without cooperation with others. However, sometimes no one could actually maximize their own profits.

Contents

Overview

Nash equilibrium is a situation in which one subject independently tries to maximize his/her own profit without collaborating with others.

Two important things about Nash equilibrium are shown below. Detailed explanation will be made in the next section.

  • Under Nash equilibrium, even if only one subject changes its own act, the subject's profit will not increase from the current situation because each subject or organization has chosen an act that maximizes their own profits while taking others' acts into consideration. Therefore, no subject or organization has an incentive to change its own acts.
  • Even though Nash equilibrium is caused as a result of each subject's act chosen to maximize its own profit, sometimes no one's profit is not actually maximized. In addition, there could be more than one combinations of subjects' acts that result in Nash equilibrium.

Nash equilibrium is named after an American mathematician John Forbes Nash Jr., who originally presented it.

Nash equilibrium and Optimal Status

Nash equilibrium is a situation in which each subject has chosen an act to maximize its own profit while taking others' acts into consideration, so it apparently looks preferable. However in fact, there could be other combinations of acts taken by both parties that both can benefit more. Or, it might be a situation in which one of the subjects can not maximize its own profit. Typical examples are as follows.

Prisoner's Dilemma

Assuming a case in which two arrested prisoners that collaborated to commit a crime are under interrogation by police officers separately. Then, let us think about an act that each prisoner would take when the police officers offer the following conditions to prisoners.

Condition
  1. Both keep silent. => Both will be imprisoned for 1 year.
  2. One keeps silent ,and the other betrays. => The prisoner that keeps silent will be imprisoned for 2 years, and the one who betrays will not be imprisoned.
  3. Both betray. => Both will be imprisoned for two years.

All possible combinations of acts that the two prisoners can take are summarized as follows. The pair (A, B) represents the duration of imprisonment sentenced to Prisoner A and B respectively.

Prisoner B Keeps Silent Prisoner B Betrays
Prisoner A Keeps Silent (1 year, 1 year) (2 years, 0)
Prisoner A Betrays (0, 2 years) (2 years, 2 years)

When Prisoner A thinks whether he should betray or not, regardless of Prisoner B's decision, choosing to betray will result in shorter imprisonment. The same thing can be said to Prisoner B. Thereby, both will choose to "betray."

This combination is Nash equilibrium. However, if both choose to keep silent, the imprisonment sentenced to them will be shorter. Thus, it is not an "optimal" situation.

Battle of the Sexes

While both a man and a woman agree on going out, the man insists on going to watch a boxing match and the woman insists on going to watch a musical. They agree on going to the same place and are trying to find a way of cooperation without communicating with each other.

The degrees of satisfaction when they choose go to watch a boxing match and a musical are supposed as follows. (M, W) indicates the degree of satisfaction of the man (M) and a woman (W) respectively. In addition, since they agreed on going out together, the degree of satisfaction when they choose different places is set to be (0,0).

Woman Boxing Woman Musical
Man Boxing (2, 1) (0, 0)
Man Musical (0, 0) (1, 2)

In this example, Nash equilibrium is caused when both go to watch a boxing match or both go to watch a musical, so there are more than one combination that cause Nash equilibrium.

One thing which these two examples have in common is that even though there are options to increase their own profits, they do not have incentives to choose such options when they act independently.

Nash Equilibrium in Business Process Optimization

When we aim to improve business process, sometime we can reach the optimal state, but sometimes we settle in a non-optimal state as described above. In other words, as a result of separate attempts for local optimization by each participant and department executing business process, the entire process is not necessarily fully optimized while each part of business process is optimized.

In order to avoid such a situation, BPM is required. BPM aims to the optimization of the entire business process.

Liable Nash Equilibrium When Executing BPM

Let us think about a case in which BPM is separately executed by each department. If business process is managed in cross-departmental manner, even though each department concerning business process can optimize flows that the department is in charge, none of them can know whether the entire business process is optimized or not. We can easily imagine a case in which cooperation among departments is not done, and as a result, the entire business process remains in unfavorable state.

This is exactly "Prisoner's dilemma," which is Nash equilibrium. Because business process in each department is optimized, no one is motivated to do global optimization.

Measures to Avoid Nash Equilibrium When Executing BPM

In order to avoid falling into non-optimal Nash equilibrium when executing BPM, all departments need to cooperate to manage business process as a whole by means of cross-departmental collaboration etc. for the sake of global optimization. In other words, unless BPM is not worked on across the company, it is not guaranteed that BPM surely takes effect. Without cross-departmental cooperation, we could easily remain in a non-optimal state and would not have any motivation to improve the situation.

An optimal state that is not Nash equilibrium (In "Prisoner's Dilemma" example, both prisoners keep silent.) is not a stable state, so we must monitor and manage the state continuously so that it will not fall into Nash equilibrium.

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