Starting 1:50 PM, 3 May 2002:
Deriving equilibrium equations and associated equations.
The modeling for three players "negotiating and bargaining" has 42 "behavioral" variables and 39 "strategic" variables.
The aifj variables: a1f2, ... a3f2, are "acceptance probabilities" and the model makes these dependent on "demand strategies" di: d1, d2, d3. Here there are 6 of the behavioral variables and only three strategic.
Acceptance behavior after the first stage of voting to accept agencies becomes differentiated by the differing pre-histories. For example, if at the first stage Player 1, acting in the style of "a1f2", succeeded in "accepting" Player 2 as his agent then what remains are Player 3 and coalition (21) (where this notation means "coalition of 2 and 1, HEADED by 2").
And now, with (3) and (21), we can have either party voting to ACCEPT the other as the controlling agent (as thus as final agent or "general agent"). So the notation is that "af21" is the probability of Player 3 voting to accept the agency control by (21) (and thus by Player 2, finally). Dually, "a21" is chosen to signify acceptance BY (21) of what there is the possibility to accept, which here is simply the agency of Player 3.
Now observe HOW MANY "acceptance" quantities (probabilities, rates) there are: There are six of aifj, six of aij, and six of afij. So 18 in all.
For the later stage acceptance probabilities there are associated "demands", one for each, controlling them so that they vary in reaction to the payoffs that can be expected by a player thus accepting another player
as agent. Thus for a12 there is d12, for af12 there is df12, etc.
But for acceptance at the first stage, when a player has alternatives
about WHICH AGENT to accept, one "demand" controls two "acceptance rates". Thus d1 (the "demand" of Player 1) controls both a1f2 and a1f3 with
Player 1 set to automatically allocate acceptance among Player 2, Player 3, and the third alternative of accepting neither of those. This leads to
n1 + a1f2 + a1f3 = 1 , which is just an equation of probabilities.
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The formulae describing the chosen (model characteristic) relation of
"demand" quantities to "acceptance" rates are listed in the subdirectories
"A12w.Af12w" and "A1f2w" of the subdirectory 4N.2002 of the directory G (for
game theory).
The payoff vector has been calculated as a function of a-quantities (rates
of acceptance) and u-quantities (utility allocations).
There are 4 distinguishable varieties of utility allocations and of each
variety there are 6 cases (6 is the permutations of three items).
A set of 4 directories is planned for these as equilibrium equations are
derived (computed) for them.
"Ujbirjk" is one of these. This, in "irjk", means that Player # i has
become the final (general) agent and thus has become enabled to allocate
the utility total realizable from cooperation (Pareto efficient cooperation).
In particular, with i, j, and k all not equal and all 1, 2, or 3, the
meaning of ujbirjk is "the quantity of utility allocated to Player j BY
Player i WHEN the situation is that i now represents all players after,
previously, Player j had become the representative of Player k. (Thus the
letters "jk" in "rjk" simply indicate that history.) (Player i here is
offering a reward to Player j for "accepting" him (i) as final agent and
the specified utility is that which Player j WILL RECEIVE provided that
Player i is elected as final agent.)
"Ukbirjk" is another directory for such computations. ukbirjk DOES NOT (!)
reward Player k for finally electing Player i as general agent and RATHER,
it rewards Player k, in effect, for allowing this situation to come about
by previously accepting Player j so that Player j could accept Player i as
final agent. (It will take the actual computations to discover what the
equilibrium values of these choices may be!)
"Ujbijrk" and "Ukbijrk" will have the calcs for other cases. Only one
equation will need to be derived in each variety and then the "permuting"
operators in the file "chiv.gen.416" will generate the other five.
For example, if the equation for the equilibrium condition applying
to u2b1r23 has been derived, say as equ2b1r23, where this is a formula to
be set = 0 to form the equation, then we can simply obtain equ3b2r31 as
equ3b2r31 = equ2b1r23//ch1
as an instruction to be used in MATHEMATICA. This is entirely equivalent to
equ3b2r31 = ch1[equ2b1r23] in MATHEMATICA.
And similarly, equ1b2r13 = ch2[equ2b1r23] because "ch2" is the operator
for interchanging players 1 and 2 while "ch1" is the operator for the cyclical permutation, 1 -> 2 -> 3 -> 1 -> 2 , etc.
(These two basic permutations generate the group of all 6 permutations
of the three items.)
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Continuing, evening of 13 May (Monday). Today, in the directory that
is called Ujbijrk I completed work to derive an equilibrium equation for
the variable called u2b12r3. This equation, as equ2b12r3 = 0, will
generalize to give six equations in all for variables of the form "ujbijrk".
(Here i, j, and k are to be 1, 2, or 3 in any order, all distinct. So there
are six possible cases of this type.
This calculation, leading to a formula for "equ2b12r3", depended on
understanding the dependence of a2f1 and a2f3 on u2b12r3. That is part of
the "reactive structure" of the game model. How Player 2 behaves at the
first stage of "acceptance elections" depends (via formulae chosen for
a2f1 and a2f3) on "d2" (which is the strategic choice of Player 2) and
on the payoff EXPECTATION of Player 2 on the condition that either
succeeds in accepting Player 1 as agent, via a2f1, or accepting Paler
3 as agent, via a2f3. These expectations first determine A2f1 and A2f3
and then form these quantities the actual behavior probabilities a2f1
and a2f3 are defined so that each of these is between 0 and 1 and so
that also their sum is between 0 and 1. (And then n2 is defined so
that n2 + a2f1 + a2f3 = 1 and n2 is the probability, at each stage
of first voting, that Player 2 votes for himself (by declining to vote
for Player 1 (via a2f1) or for Player 3 (via a2f3)).
More started 26 May 2002, continued 14 June:
A few days ago, after finding the error in the payoff functions (vector)
("ppvd") caused by the effective absence of any contribution from the
coalition values b1, b2, and b3 (for (2,3), (1,3), and (1,2) coalitions),
I also realized that there was a failure to properly consider or take into
account the effect of "reactions" in calculation of the equilibrium conditions relating to aij (for dij) and afij (for dfij).
The complication is that, analogously with u2b12r3 (as mentioned above),
the quantities a21 and af21 affect the evaluation of the "utility prospect"
as seen by Player 1 when he/she considers whether or not it is appropriate,
considering his/her "first stage demand" d1, to "accept" Player 2 as agent
as the game goes into the second stage. The "utility prospect" naturally
depends on what Player 2 and Player 3 will do in the second stage if Player 1
has given agency power to Player 2 and these behavior patterns are described
by a21f3 (or a21) and a3f21 (or af21). (And also relevant to the evaluation
of Player 1's relevant utility prospect in relation to his a1f2 relating option are, of course, u1b21r3 and u1b3r21.)
Further, since the formulae for a1f2 and a1f3 are structured so as to make them never add to more than 1 by making a1f2=A1f2/(1+A1f2+A1f3) and
a1f3 = A1f3/(1+A1f2+A1f3), it happens that BOTH a1f3 and a1f2 depend on the utility prospect (seen by Player 1 when considering whether to "accept" Player 2 as agent (in the first stage of acceptance elections) although only A1f2 (of the formulae A1f2 and A1f3) depends on this specific utility prospect.
So for deriving the correct equilibrium conditions relating to d21 (via a21) and df21 (via af21) we must take into account that a1f2 and a1f3 behave as functions of a21 and af21 in this context. (This makes them also functions of d21 and df21 through the direct dependence of a21 on d21 and af21 on df21.)
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Regarding the correction of the original "ppvd" calculation of the payoff
vector formula, this was done as "ppvdd" (and forms using ni variables and/or
other non-final parameters or probability quantities) were calculated as
ppvaa, ppvbb, and ppvcc.
Later "U" was independently calculated (by AK). This became U4 after some
non-final parameters were replaced by appropriate systematic substitutions
and/or re-naming. U4 turned out to be longer than ppvdd, but equivalent. And
later U4 was simplified in form to U6 which now seems to be a lot shorter
than either of ppvdd and U4. (These issues of shortness may ultimately affect
the speed of calculations involving the derived equilibrium condition equations.)