Monday, November 1, 2010

Nash Equilibrium Part II

Here we are back with Mr John Nash and his equilibrium, called conveniently enough: the "Nash Equilibrium".  If you missed Part I, you can get it here.

Today, we take a look at what Alan Blinder did to move it along. Our spin is on project management, different from what Nash and Blinder used as the context, but nevertheless, the Nash Equilibrium shows up in project situations more often than we might imagine.

Equilibrium refers to the idea that the game ends with everyone winning something, or no one losing everything. In other words, a state of balance exists between the parties when the game ends. Thus, equilibrium applies to non-zero-sum outcomes.

Just to review:
Just to review Part I of this series: the Nash Equilibrium comes out of 'game theory' that is itself a study of how wary adversaries interact....most often in non-zero-sum situations in which they find themselves mutually involved. It is presumed that there is a sequence of moves by the parties in accordance with some protocol.

Game theory is somewhat related to the more familiar 'utility theory' that describes how one or more persons' attitude is affected by the persons' relationship to [an inanimate] condition, risk, or event.

The Blinder Matrix
Here's what Blinder gave us in the so-called 'payoff' matrix or the "Blinder Matrix":

The setup: Two parties face a situation. A resolution is needed. Each makes decisions independently. Each has similar tools and capabilities and preferences for an outcome, but they weigh each differently. What's most important to one is not so important to the other. And each has a pretty good idea of the other's priorities, tools, and capabilities.

Project Game
For the project game, let's assume that both cost and time are of importance to both the project team and to the stakeholder community, but each sets a different priority.  We'll assume scope is constant.  In the first figure below, we see a partially completed "Blinder Matrix".  Stakeholder preferences are in the upper side triangle of each square.  You can see that there are four combinations of two preferences between two parties.

"cost/cost; cost/time; time/cost; and time/time"

We see that the stakeholders most favored preference is time and it is given the number 1 to indicate it's first rank.  Cost is the least favored preference by the stakeholders [after all, project cost usually doesn't show up on stakeholder P&L].



Now, let's fill in the project preferences. In this 'game', project preferences are opposite the stakeholders; the PM values cost first.


Play the game
So, with the preferences established, let's play the game. Each party knows the other's preferences and thinks they know the other's strategy to react to the other's move. Each party is independent; each party is not looking for an optimum solution; they are looking for a stable solution that they can both live with.

The next figure shows the strategy reactions by preference. For example, the upper right square, time/cost, is second rank for project after its first ranked preference time/time because at least the project is still focused on time. The project's third ranked preference is cost/time; the time 'agenda' is shifted to the stakeholder but that's better for the project than cost/cost that has no time preference.


Game moves
Let's say the stakeholder decides to make the first move: if the stakeholder chooses Time as their first move, then the project is not going to respond with time since a time/time situation ranks 4 for the PM; if the stakeholders goes with Cost, then the project will jump on that and the upper square situation will result with everyone lined up on cost; this ranks 4 with the stakeholders so stakeholders won't start with cost.

Now, here's the equilibrium: there's no way to reach the upper right square with the stakeholder making the first move with Time.  The stakeholder will go first with their higher preference, Time, along the lower row. If PM responds with time, then the game is an unstable 1/4 state; but we expect    the PM will respond with their higher preference, Cost, and the game equalizes on the time/cost 3/3 square in the lower left, which is 3rd preference--a clear sub-optimization--but nevertheless stable!

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