Anyone want to try to figure this out? It's a Game theory problem.

Greetings,
I need help with this Game theory Formula. I have been working with this model for a while now (have a spreadsheet example that took months to put together, it had a lot of input data) but I'm not sure if I understand it correctly. I would like to see what other people get with the following input data:

c=[100,100,50,25]
s=[100,100,10,50]
x=[100,0,50,60]

The Formula in question is on pages 20-22 in this Journal Article https://oficiodesociologo.files.wor...-bueno-de-mesquita-s-group-decision-model.pdf
https://oficiodesociologo.files.wor...-bueno-de-mesquita-s-group-decision-model.pdf

On page 19 of the following pdf is the Weighted Mean Position, the rudimentary version of the model above. This model I understand just fine, below is an example of said formula.

http://irworkshop.sites.yale.edu/si...odel for Predicting Policy ChoicesREvised.pdf


What this model measures is bargaining position and resulting outcome position of negotiations after several rounds. In this case 100 is Bob's advocated position(say Bob wants a funding bill to pass Congress), 0 is Bill's advocated position(Bill does not want the bill to pass). Jane and Sarah are somewhere in the middle on the position spectrum. Salience measures how much clout you invest in the issue at hand. Power is applied influence. 0 salience means no investment in the issue meaning Jane isn't doing much to get what she wants. 100 salience means complete investment in the issue.

Here is the result of the negotiation: 1100000/21750=50.574713


Sarah nudged the results slightly, otherwise it would have been exactly 50.
I set the input so the people have a fairly equal influence distribution so the the resulting position should be close to 50 (the outcome in the middle).
The idea is to try to predict alliances and how an individual's bargaining position will change per round, based on the decisions and power of the other people involved in the negotiation. It measures this in part by a decision tree system called utilities. On page 8 is a graph of utilities. There are five utilities and it is basically a measure of the usefulness of certain decisions. For example: Sarah has to decide the utility of doing nothing or bargaining. The decision then branches out into two options: she can gain or lose. Utilities can be summarized as follows 1=win, 2=lose, 3=silently win, 4=silently lose 5=status quo.

Here is the formula in the Scholz document: the x{i} -x{j} notation means x subscript i -x subscript j. The i, j, k subscripts are a way of telling whose position it is. x{i} is another way of saying 'Bob's Position.

All calculations in this total formula is just to determine the new position for 'i'.

1. Given i=1,2,....., n actors, initial positions for each actor x(index i)(t=0),c,s, and number of rounds=t c=clout or influence or unapplied power, s=salience or investment in issue, x=position on spectrum

2. let r(index i)=1 Note:This is risk propensity, it estimates how risky your position is in the game. You have to set it to 1 initially because you haven't calculated this value yet. After you have done so, then plug it into the utility calculations the second time you go through the steps again. I will from now on denote index with the following brackets:{}

3. Calculate pairwise votes: V^jk=SUM [i to n with i=1] ((|x({i}-x{k}|-|x{i}-x{j}|)/|x{max}-x{min}|) Then find the maximum value that corresponds to the Condercet winner position or median=u This is talking about a condercet voting system, in which a winner of an election is determined by who wins the most head to head matches. It is also called the weighted median position. I have another way of calculating this with a table which I can present if you would like. Or you could just use the weighted mean position instead for this part of the calculations as this part of the formula is trying to determine the perceived middle position on the issue spectrum. Also that 'exponent jk' does not mean raise to a power, it just means 'j vs k'.

4. Calculate Basic Utilities for i:
U^si=2-4(0.5-0.5|(x{i}-x{j})/(x{max}-x{min})|)^r{i} This is the Challenge opponent and win result. Notice the sign between the two 0.5's? It will change for its inverse calculation. This utility equation should also have an absolute value of x{max}-x{min}. This is the range of positions on the issue spectrum.

U^fi=2-4(0.5+0.5|(x{i}-x{j})/(x{max}-x{min})|)^r{i} This is the challenge opponent-bad result.

U^bi=2-4(0.5-0.25(|x{i}-u|+|x{i}-x{j}|/|x{max}-x{min}|)^r{i} This is the not challenge opponent-lucky result.

U^wi=2-4(0.5+0.25(|x{i}-u|+|x{i}-x{j}|/|x{max}-x{min}|)^r{i} This is the not challenge opponent decision-unlucky result.

U^sq=2-4(0.5)^r{i} This is the status quo Utility, notice the 0.5 for uncertainty. The utilities in total should be similar to this: -2<U^fi<U^wi<0<U^bi<U^si<2 Of course be mindful that Risk propensity exponent will alter this order.

5. Calculate probabilities:
P{i vs i}=SUM(k if argument>0) (c{k}s{k})(|x{k}-x{j}|-|x{k}-x{i}|)/SUM(k to n with k=1) (c{k}s{k}|(|x{k}-x{j}|-|x{k}-x{i}|)| This is actor 'i' calculating it's own chances of winning. 'i' also calculates the chances of its opponents winning as well.

6. Let Q=0.5 (or 1.0) This is the probability of the status quo. Check page 11 of scholz paper for more details.

7.Calculate:
Ei(Uij)=s{j}(P{i vs i}U^si+(1-P{i vs i})U^fi)+(1-s{j})U^si-QUsq-(1-Q)(TUbi+(1-T)U^wi) 'i' is calculating the expected utility of a contest with 'j'. Hence why salience of 'j' is factored in here. T is the probability of i's welfare improving if j moves on the issue spectrum. 0=bad 1=good from i's perspective. There are pictures of this on page 11 of the scholz paper.

Ej(Uji)=s{j}(P{i vs i}U^sj+(1-P{i vs j})U^fj)+(1-s{j})U^sj-QUsq-(1-Q)(TUbj+(1-T)U^wj)
There are four expected utilities to calculate by the way. Two from i's perspective and two from j's perspective.

If Second Pass go to step 11.

8. Calculate:
R{i}=2SUM(j to n, j=1, and j=/=i) Ei(Uji)-max i(SUM(j to n, j=1, and j=/=i) Ei(Uji))-min i(SUM(j to n, j=1, and j=/=i) Ei(Uji))/max i(SUM(j to n, j=1, and j=/=i) Ei(Uji))-min i(SUM(j to n, j=1, and j=/=i) Ei(Uji))
This looks like it can be simplified like this R{i}=2SUM(j to n, j=1, and j=/=i) Ei(Uji)-1
This is further explained on page 12 of scholz. This is basically i's definition of security: how much expected utility do the adversaries have of challenging i. It should be in this range: -1<R<+1
9. Calculate:
r{i}=(1-R{i}/3)/(1+R{i}/3)
This is i's risk propensity. You will no go to step 4 and raise the utilities by this exponent.

10. Go to step 4, using calculated values of r{i}
11. Determine new position decisions x, based on rules in Section 5 for octant of Eij(i) versus Eji(j)
Page 13 of scholz. Basically i will now make a decision on what to do. It's options are Conflict, Compromise, Capitulate, Stalemate.
Conflict is when both i and j's expected utility is positive. In other words, both sides think they can win a confrontation. Ei(Uij)>0 and Ej(Uji>0
Compromise is when one side knows they will lose, but it won't be easy for their opponent, so they can maybe make a deal where the loser moves part way to the winner. Ei(Uij)>0 and Ej(Uji)<0 and |Ei(Uij)|>|Ej(Uji)| OR Ei(Uij)<0 and Ej(Uji)>0 and |Ei(Uij)|<|Ej(Uji)|
Compromise formula for octant Compromise+

New x=(x{i}-x{j})|Ej(Uji)/Ei(Uij)|
Or if in Octant Compromise -
New x=(x{i}-x{j})|Ei(Uij)/Ej(Uji)|
Capitulate is when one side knows they will lose dramatically and the loser will have to accept the stronger actor's position. Ei(Uij>0 and Ej(Uji)<0 and |Ei(Uij)|<|Ej(Uji)| OR Ei(Uij)<0 and Ej(Uji)>0 and |Ei(Uij)|>|Ej(Uji)|
Stalemate is when neither side sees a gain from confronting each other. Ei(Uij)<0 and Ej(Uji<0 Neither expects to move from their current position
12. Increment the rounds, t=t+1
13. If t=tau then stop.

Picture(Note the L, J values in the following image are wrong)

Thanks,

Clarification.:

The picture input is different but the results using a variation of the input data in the above example (Sarah is excluded in this variation) is c=[100,100,50] s=[100,100,10] x=[100,0,50] should be something along this line: V=500,000 U=1 M=-1 Utility L=1 Utility J=-1 Z=0 p=-1 Ei(Uij)=-39 Ei(Uji)=0.75

Picture(Note the L, J values in the following image are wrong) View attachment 4461

Thanks,

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