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Designing Outcomes (Applied game theory – Part 2)

February 4, 2010

It seems natural to extend the basic idea from the Predictioneer’s Game introduced in the last post in two ways.

First, what happens if subsets of the group meet and some of those parties sway others or reach compromises in order to form coalitions?

Second, taking this idea further, can we investigate a variety of decision processes to find an optimal process for a desired outcome and design coalitions in order to reach a desired outcome?  In terms of the example situation,  this is the problem of finding coalitions that will drive a desired result from those we predicted last time.

Again, here is the prediction from the last post:

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):        76.4
Position (balance of power):    71.7
TABLE 1

To answer the first question, we need to build a simple model that replaces a coalition with a single new entity. In the model, the choice was to add the Influence of the parties of a coalition.  The new position was the weighted average of the positions as described before.  For the Salience, we need to reflect the idea that the most interested party in a coalition will drive the others and use their influence to support the new positions but with some skewing toward their Salience.  To model this simple, replace the new entity’s salience with 20% of the average Salience + 80% of the max Salience.  Many other choices can be made and explored using the tools here.

To answer the second question, we need a way to generate all of the possible coalitions.  This amounts to generating a set partitioning of all the parties.  TABLE 2 shows a simple example with only 4 parties.

Parties    :     ['a', 'b', 'c', 'd']
<<<  15 Partitions  >>>
-----------------------------
[['a', 'b', 'c', 'd']]
[['a', 'b', 'c'], ['d']]
[['a', 'b', 'd'], ['c']]
[['a', 'b'], ['c', 'd']]
[['a', 'b'], ['c'], ['d']]
[['a', 'c', 'd'], ['b']]
[['a', 'c'], ['b', 'd']]
[['a', 'c'], ['b'], ['d']]
[['a', 'd'], ['b', 'c']]
[['a'], ['b', 'c', 'd']]
[['a'], ['b', 'c'], ['d']]
[['a', 'd'], ['b'], ['c']]
[['a'], ['b', 'd'], ['c']]
[['a'], ['b'], ['c', 'd']]
[['a'], ['b'], ['c'], ['d']]
TABLE 2

For 10 parties, there are 115,974 games to play out.  (See Donald Knuth’s The Art of Computer Programming Vol 4 F3).  You can download output of all 115,974 games.  The interesting games are the extremes: the game that results in the maximum position and the one that results in the minimum position.

Game A
maxPos : 83.22147
party                           :    Pos        Inf(Norm)    Sal
----------------------------------------------------------------
eng                             :    60        0.07547        99
_d1+d2_                         :    64        0.18868        95
_er+me_                         :    74        0.16981        83
_adv+ctcust+ctsal+inv+legal_    :    97        0.56604        77
-----------------------------------------------------------------
Position (weighted avg):    83.2
Position (balance of power):    77.1
(Game # : 2089)
Game B
minPos : 66.27527
party                    :    Pos        Inf(Norm)    Sal
------------------------------------------------------------
_ctcust+ctsal+d2+inv_    :    32        0.32075        65
_d1+eng+me_              :    72        0.37736        99
_er+legal_               :    98        0.15094        80
adv                      :    100       0.15094        20
------------------------------------------------------------
Position (weighted avg):    66.3
Position (balance of power):    51.2
(Game # : 47615)
TABLE 3

In Game A, a coalition of d1+d2 and er+me results in the most extreme high-number position. The final position realized was 100 and these two coalitions were instrumental in the outcome.  To design for to most extreme high position, these are the coalitions to nurture.

In this case, however, d2, the biggest stakeholder in the low-number position, would have achieved more of their goals had they designed the decision process around the coalitions shown in Game B.  By building a relatively strong coalition around ctcust+ctsal+d2+inv and a diffusive coalition around er+legal, a much lower position is achieved.

There is a second hint here too.  If the decision process could be more focused on Balance of Power rather than a salience/influence weighted average, a position of around 50 might be reached.   This might be done, for example, by getting all the parties to agree to a final vote with 1 vote per person, winning position takes all votes, then voting in rounds sequentially through the coalitions of Game B top to bottom.

Download the Python model modules: MultiPlayerGame.py MultiPlayerNegotiationsModel.py Partitioner.py

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