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Applied game theory – Part 1

February 1, 2010

Bruce Bueno De Mesquita’s Predictioneer’s Game describes work by De Mesquita’s team of applied game theorists and his students to business and political negotiations and decision making.   Predictioneer’s Game covers some of the very basic ides, but De Mesquita leaves nearly all of the details of his models a mystery.

I recently had the opportunity to observe a business negotiation between about a dozen parties that seemed ripe for PG analysis. Among the basic ideas explained in the book are the basic inputs used by De Mesquita’s models and how to make estimates of “game” outcomes.  In a future post the model here will be expanded to model more of the details of interactions between the parties and sub-groups. As the book contains few hints to how De Mesquita does this, those models will be striking off on our own.

But first, back to the negotiations and simple estimates of the outcome. All of the parties below are actual people involved in the negotiation.  Negotiations covered about 2 months and are completed with respect to these positions.

The object is to calculate two estimates of the outcome of the negotiations following Predictioneer’s Game. Here are the steps:

(1) Identify the stake-holding parties involved in the negotiations.  Anyone who takes a definite position and has some stake in the outcome should be included.  This model includes only the most obvious players.

Party     [Position, Influence, Salience]
-----------------------------------------
    data = {
        'd2':       [ 15,  20,  80],
        'ctcust':   [ 35,  50,   5],
        'ctsal':    [ 35,  20,   5],
        'er':       [ 50,  10,  20],
        'eng':      [ 60,  40,  99],
        'd1':       [ 75,  80,  99],
        'me':       [ 75,  80,  99],
        'adv':      [100,  80,  20],
        'inv':      [100,  80,   5],
        'legal':    [100,  70,  95]
    	}
     		TABLE 1
(2) Describe the continuum of positions held by the parties.  This requires research and disciplined appraisal of what you know about the parties and what they have said and done regarding their position.  In this case, the parties are ordered along the line of positions 0 to 100.  The first column of numbers in the table above are the numbers I recorded in my notebook during the negotiations at the time I was reading PG.  (I would give different scores today, but that would be hindsight, not prediction.) 0 equates to one of the parties taking full control of future decision making and business rewards. At the other end of the scale 100 corresponds to decisions in the hands of a subgroup of the original parties.  Negotiations are centered around a document proposing a outcome position at approximately 70.

(3) Describe the relative influence of the parties.  Below, the influence will be normalized to add up to 100%, so just choose a scale that is convenient to get the relative numbers right.  In the example, I estimated the influence of each party at the same time as I estimated positions.

(4) Now it is time to the estimate cost/benefit to each party.  Salience is a score starting with ambivalence=0 moving up to everything there is to gain or loose is staked on the outcome scoring 100.

Predictioneer’s Game describes students and analysts researching positions and salience for political problems through interviews, CIA analyst records, newspapers, eye-witness reports etc.  These steps could take a great deal of work.  But also some discipline to hear and interpret what people are trying to accomplish.

Time to estimate outcomes. The first estimate is the weighted average position given by

\bar{P} = \frac{\sum_{p} Salience \times Influence \times Position}{\sum_{p} Salience \times Influence}

Plugging in the numbers above gives a position just above parties d1 and me at 76.4. See Table 2 below.

The second estimate of outcome is the point along the position line where the cumulative total influence is 50%.  That is, there is as much influence pulling the final position to the left as to the right.  In this case the the balance-of-power point is at position 71.7.

The results are summarized on the table below.  Influence has been normalized to add up to 100%.

Party   :       Pos             Inf(Norm)       Sal
------------------------------------------------------------
d2      :       15              0.03774         80
ctcust  :       35              0.09434         5
ctsal   :       35              0.03774         5
er      :       50              0.01887         20
eng     :       60              0.07547         99
me      :       75              0.15094         99
d1      :       75              0.15094         99
adv     :       100             0.15094         20
legal   :       100             0.13208         95
inv     :       100             0.15094         5
------------------------------------------------------------
Position (weighted avg):        76.37
Position (balance of power):    71.72

		TABLE 2

Both of these predictions give a position that was discussed as a plausible compromise between the more extreme positions.

This is not how the negotiations worked out.  The final outcome was at position 100.  In fact, the middle ground evaporated about 7 days after I first made this calculation.  This situation moved very quickly to position 100 with everyone from position 50 and above accepting the 100 was the only possible outcome and the majority of the influence below position 50 also insisting that, although undesirable, 100 was the only possible outcome.

I am fairly confident that any proposal from the influential parties in the positions below 50 that landed at or slightly above 50 would have lead to further negotiations and resolution.  In this case, some of the parties decided not to fight for their positions directly. This means I overestimated the salience of their positions?

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