In 1959, Lloyd Shapley wrote a short paper on games with vector payoffs. He analyzed zero-sum matrix games. Here, we extend Shapley's equilibrium concept to general games with vector payoffs, introduce an organizational interpretation of the concept, elaborate the relationship of the original concept to another equilibrium concept where each player can be viewed as running a bargaining game among internal ‘factions,'' and finally comment upon its relationship to the concept of party unanimity Nash equilibrium (PUNE).

John Roemer

English

Research Papers in Economics

Games with vector−valued payoffs and their application to
competition between organizations 
John Roemer
Yale Univ
Abstract
In 1959, Lloyd Shapley wrote a short paper on games with vector payoffs. He analyzed
zero−sum matrix games. Here, we extend Shapley’s equilibrium concept to general games
with vector payoffs, introduce an organizational interpretation of the concept, elaborate the
relationship of the original concept to another equilibrium concept where each player can be
viewed as running a bargaining game among internal ‘factions,’ and finally comment upon
its relationship to the concept of party unanimity Nash equilibrium (PUNE).
Citation: Roemer, John, (2005) "Games with vector−valued payoffs and their application to competition between
organizations." Economics Bulletin, Vol. 3, No. 16 pp. 1−13
Submitted: March 21, 2005.  Accepted: March 25, 2005.
URL: http://www.economicsbulletin.com/2005/volume3/EB−05C70014A.pdfMarch 10, 2005
“Games with vector-valued payoff functions and their application to competition between
organizations
1”
by
John E. Roemer
Yale University
1. Introduction
In 1959, Lloyd Shapley wrote a short paper on games with vector payoffs.  He
analyzed zero-sum matrix games.   Here, we extend Shapley’s equilibrium concept to
general games with vector payoffs, introduce an organizational interpretation of the
concept, elaborate the relationship of the original concept to another equilibrium concept
where each player  can be viewed as running a bargaining game among internal
‘factions,’ and finally comment upon its relationship to the concept of party unanimity
Nash equilibrium (PUNE).
2. Games with vector payoffs
Consider a ‘game’ with n players, each of whom has several goals.  For the sake
of simplicity in exposition only, we suppose that n=2, each player has two goals, and that
they share a common strategy space S.  Denote by   U
a :S × S → ° and U
b :S × S → ° 
two payoff functions for Player One, and by   V
a :S × S → ° and V
b :S × S → ° two
payoff functions for Player Two.   We will speak of the players’ a- and b-goals.
Players are unable to assign weights to these two goals – they hence do not
possess complete preference orders over S × S.   They are limited to making comparisons
                                                   
1 I am grateful to Michel LeBreton for alerting me to the existence of Shapley’s paper.2
by dominance only, in the following sense.   A best response by Player One to a strategy
s2 by Player Two is a strategy s1 such that:
(/ ∃s ∈S)(U
a(s,s2),U
b(s,s2)) ≥ (U
a(s1,s2),U
b(s1,s2)) (1)
where the vector inequality ≥ means at least one component of the left-hand side is
strictly larger than its counterpart, and none is smaller.  Thus, define:
B
1(s2) = {s1 ∈S |  statement (1) holds}.
Define B
2(s1) in like manner.
The tuple (S,U
a,U
b,V
a,V
b) will be called a game with vector payoffs.
Definition.   A strategy pair (s1,s2)∈S × S is a vv-equilibrium of the game
(S,U
a,U
b,V
a,V
b) if s1 ∈B
1(s2) and s2 ∈B
2(s1)  .
Let α,β ∈(0,1) and consider the standard game whose payoff functions for the
two players are:
æ
1α(s1,s2) = αU
a(s1,s2)+ (1−α)U
b(s1,s2),
æ
2β(s1,s2) = βV
a(s1,s2)+ (1− β)V
b(s1,s2).
Denote this game by G
(α,β).  If U
a,U
b,V
a, and V
b are concave and continuous then
G
(α,β) possesses a (standard) Nash equilibrium, call it (s1(α,β),s2(α,β)).  But this is also
a vv-equilibrium of G.  For suppose there were a strategy s such that
(U
a(s,s2(α,β)),U
b(s,s2(α,β))) ≥ (U
a(s1(α,β),s2(α,β)),U
b(s1(α,β),s2(α,β))).
Then, since α ≠ 0 ≠1−α, we have æ
1α(s,s2(α,β)) > æ
1α(s1(α,β),s2(α,β)), an
impossibility.
Thus we have immediately a two dimensional manifold of vv-equilibria.  (One
should perhaps worry about whether these are distinct.)3
Are there other vv-equilibria?  Possibly – but not very many.  Define the utility
possibilities sets:
 Φ
1(s2) = {(u
a,u
b)∈°
2 |(∃s ∈S)((u
a,u
b) = (U
a(s,s2),U
b(s,s2))},
and Φ
2(s1) in like manner.  Assume free disposability – that is, Φ
2(s1) and Φ
1(s2) are
comprehensive sets in  °
2  for all s1,s2.  Then these sets are convex.  For let s1, ˆ s1,s2  be
arbitrary.Then
(U
a(λs1 + (1− λ)ˆ s1,s2),U
b(λs1 + (1− λ)ˆ s1,s2)) ≥ λ(U
a(s1,s2),U
b(s1,s2))+ (1− λ)(U
a(ˆ s1,s2),U
b(ˆ s1,s2))
by concavity.  Hence, by free disposability, the point on the r.h.s. of the above inequality
lies in Φ
1(s2), which demonstrates that it is as convex set.
Now let (s1,s2) be a vv-equilibrium of G.  Consider the convex set
P
1(s1,s2) = (U
a(s1,s2),U
b(s1,s2))− Φ
1(s2).
Let  Ω = {(x,y)∈°
2 | x ≤ 0,y ≤ 0}  be the non-negative quadrant of  °
2 .   Then
P
1(s1,s2)∩Ω = {(0,0)};
if any other point were in the intersection of these two sets, then s1 would not be a vv-
best response to s2 .  Therefore, by the separating hyperplane theorem, there is a non-zero
vector  p ∈°
2 such that
p⋅Ω ≤ 0, p⋅P
1(s1,s2) ≥ 0.
By the first inequality, we must have  p ≥ 0 .  Therefore, ignoring scale,  there is a number
α ∈[0,1] such that  p = (α,1−α).
In like manner, there is a number β ∈[0,1] such that (β,1− β)⋅P
2(s1,s2) ≥ 0 ,
where4
P
2(s1,s2) = (V
a(s1,s2),V
b(s1,s2))− Φ
2(s1).
Because  (α,1−α)⋅P
1(s1,s2) ≥ 0 , it immediately follows that
s1 maximizes æ
1α(s,s2); in like manner, s2 maximizes æ
2β(s1,s).  Thus, (s1,s2) is a
Nash equilibrium of the standard game G
(α,β).   Summarizing:
Theorem 1.   Let U
a,U
b,V
a,V
b be continuous, concave payoff functions, let free
disposability hold in the vv game G = (S,U
a,U
b,V
a,V
b), and let S be convex.  Then:
(1) For every (α,β)∈(0,1)
2 the Nash equilibria of the game G
(α,β) are vv-
equilibria of G;
(2) If (s1,s2) is a vv-equilibrium of G, then there exists (α,β)∈[0,1]
2  such that
(s1,s2) is a Nash equilibrium of G
(α,β).
Clearly, the theorem generalizes to any number of players each with any number
of goals.  If there are n players and player i has ni  goals, then the dimension of the
equilibrium manifold is  (gi ∑ −1) = gi ∑ − n.
3.  A bargaining interpretation
We motivated the vv game by describing individuals with multiple goals.   In
applications, it will often by the case that each player is an organization, whose members
have different goals.   Call the set of members in an organization of share the same goal a
faction.   The organizations might be political parties, firms, or trade unions.  In this5
context, one might propose that factions would bargain with each other, in the face of a
strategy put forth by the opposition organization.
This motivates the following set-up.  Let here be two organizations, each with an
a and b faction.  Each faction possesses von Neumann-Morgenstern preferences over
lotteries on S × S.  Let   U
a :S × S → °, U
b :S × S → °  be vNM utility functions
representing these preferences for the a and b factions in Organization 1, and let
  V
a :S × S → ° ,  V
b :S × S → ° represent the vNM preferences of the factions in
Organization 2.   Suppose there are ‘status quo’ or ‘impasse’ strategies s1
0 and s2
0  for the
two organizations;  if the factions in an organization cannot reach agreement in their
bargaining, then the organization plays the impasse strategy.    We also take as data a
number α ∈[0,1], called the bargaining power of faction a in Organization 1, and a
number β ∈[0,1], called the bargaining power of faction a in Organization 2.   We define
an organizational game as a tuple OG = (S,U
a,U
b,V
a,V
b,s1
0,s2
0,α,β).   We define the
tuple (S,U
a,U
b,V
a,V
b,s1
0,s2
0) to be an organizational game form.
We define:
Definition.  A Nash bargaining solution equilibrium of the organization game OG is a
pair of strategies (s1,s2)such that
s1 ∈argmax
S
α log(U
a(s,s2)−U
a(s1
0,s2))+ (1−α)log(U
b(s,s2)−U
b(s1
0,s2))
and
s2 ∈argmax
S
βlog(V
a(s1,s)−V
a(s1,s2
0))+ (1− β)log(V
b(s1,s)−V
b(s1,s2
0)).
We say a Nash bargaining solution equilibrium is non-trivial if the utility of all
four factions at the equilibrium strategy pair is strictly greater than it would be if each6
organization played its impasse strategy, facing the other organization’s equilibrium
strategy.   (I.e., each faction strictly gains from cooperating with its partner faction.)
The definition says that,  given the strategy of Organization 2,  strategy s1 solves
a Nash bargaining game between the factions of Organization 1  (i.e., it maximizes the
appropriate weighted product of the utility gains from the impasse point of the two
factions), and given the strategy of Organization 1, strategy s2  solves the Nash
bargaining game between the factions of Organization 2.
We have:
Theorem 2. Let (S,U
a,U
b,V
a,V
b,s1
0,s2
0) be an organizational game form such that, for
every choice of s2 and s1, the four functions
log(U
a(⋅,s2)−U
a(s1
0,s2)),log(U
b(⋅,s2)−U
b(s1
0,s2)),
log(V
a(s1,⋅)−V
a(s1,s2
0)), and log(V
b(s1,⋅)−V
b(s1,s2
0))
 are concave on the strategy
domains where they are defined  (i.e., where the arguments of the log functions are
positive).  Then for every (α,β)∈[0,1]
2 , there exists a Nash bargaining solution
equilibrium for the organization game (S,U
a,U
b,V
a,V
b,s1
0,s2
0,α,β).
Proof:
Given (α,β)∈[0,1]
2 .   Define the payoff functions:
æ
1(s1,s2) =
(U
a(s1,s2)−U
a(s1
0,s2))
α(U
b(s1,s2)−U
b(s1
0,s2))
1−α, 
               if U
a(s1,s2) ≥U
a(s1
0,s2) and U
b(s1,s2) ≥U
b(s1
0,s2)  
0, otherwise





æ
2(s1,s2) =
(V
a(s1,s2)−V
a(s1,s2
0))
β(V
b(s1,s2)−V
b(s1,s2
0))
1−β, 
               if V
a(s1,s2) ≥V
a(s1,s2
0) and V
b(s1,s2) ≥V
b(s1,s2
0)  
0, otherwise




7
By the log concavity premise, the conditional payoff functions æ
1(⋅,s2) and æ
2(s1,⋅) are
quasi-concave for any choice of s1 and s2 .   It therefore follows, by the standard Nash
equilibrium existence theorem, that a Nash equilibrium exists for the game (S,æ
1,æ
2).
But this is Nash bargaining solution equilibrium for the organizational game
(S,U
a,U
b,V
a,V
b,s1
0,s2
0,α,β).     n
We next exhibit the relationship between Nash bargaining solution equilibrium
and vv equilibrium.
Theorem 3. Let U
a,U
b,V
a,V
b be payoff functions on the domain S × S, and let
s1
0 and s2
0  be impasse strategies for organizations 1 and 2.  Then:
(1) Let (α,β)∈[0,1]
2  and let (s1,s2) be a Nash bargaining solution equilibrium for the
organizational game associated with these data.  Then (s1,s2) is a vv-equilibrium of the
vv game G = (S,U
a,U
b,V
a,V
b).
(2) Let the four functions log(U
a(⋅,s2)−U
a(s1
0,s2)), etc., be concave on the domains
where they are defined.  Let (s1,s2) be a vv-equilibrium of the game
G = (S,U
a,U
b,V
a,V
b) such that:
(2a)
U
a(s1,s2) >U
a(s1
0,s2)
U
b(s1,s2) >U
b(s1
0,s2)
V
a(s1,s2) >V
a(s1,s2
0)
V
b(s1,s2) >V
b(s1,s2
0)







Then there is an ordered pair (α,β)∈[0,1]
2  such that (s1,s2) is a Nash bargaining
solution equilibrium of the organizational game (S,U
a,U
b,V
a,V
b,s1
0,s2
0,α,β).
Proof:8
Part (1).  Given (α,β) and (s1,s2) according to the premise.  Define K
1 =U
b(s1,s2).  We
know that the strategy s1 solves the program
max
S U
a(s,s2)
s.t.U
b(s,s2) ≥ K
1;
this uses the fact that (s1,s2) is a non-trivial Nash bargaining solution equilibrium.  But
this means that s1 ∈P
1(s2).    In like manner, s2 ∈P
2(s1).  The claim is proved.
Part (2).  Let (s1,s2) be a vv-equilibrium which satisfies the premise (2a).  Define the sets
 A = {(x,y)∈°
2 |(∃s ∈S)((x,y) = (log(U
a(s,s2)−U
a(s1
0,s2)),log(U
b(s,s2)−U
b(s1
0,s2)))},
 B = {(x,y)∈°
2 |(∃s ∈S)((x,y) = (log(V
a(s1,s)−V
a(s1,s2
0)),log(V
b(s1,s)−V
b(s1,s2
0)))} .
By inequalities (2a), these sets are non-empty.  By the log concavity assumption, they are
convex sets.   Since (s1,s2) is a vv-equilibrium, the point (x(s1),y(s1))∈A generated by
the strategy s1 lies on A’s northeast boundary.  Therefore, there exists a supporting line
for A containing this point whose normal vector is non-negative.   It follows that there is a
number 0 ≤α ≤1 such that
s1 ∈argmax
S
α log(U
a(s,s2)−U
a(s1
0,s2))+ (1−α)log(U
b(s,s2)−U
b(s1
0,s2)).
In like manner, there is a number 0 ≤ β ≤1 such that
s2 ∈argmax
S
βlog(V
a(s1,s)−V
a(s1,s2
0))+ (1−α)log(V
b(s1,s)−V
b(s1,s2
0)).
It follows that (s1,s2) is a Nash bargaining solution equilibrium of the constructed
organizational game.   n9
Theorem 3 tells us that, under suitable conditions, we can interpret vv equilibria
as solutions of ‘hypothetical’ organizational games in which factions representing
different ‘interests’ of the individual players are bargaining with each other.
  There are, it seems to me, two uses of this theorem.   The first is in the case
where the players are actually individual persons, who have multiple interests, and who
have incomplete preference orders over S × S, because they are unable to aggregate their
multiple interests into a coherent preference order.    Here, the interpretation is that any
equilibrium can be rationalized as an equilibrium of an associated organizational game,
where each interest is represented by a faction, and the factions bargain à la Nash.  In
other words, Nash bargaining is all that one needs to resolve the incompleteness of
preferences generated by multiple interests.   The second interpretation is in the case
where the players are actually organizations with factions.  The interpretation is in this
case that, regardless of what the actual bargaining process between the factions is, as long
as bargaining is efficient (i.e., produces a strategy from which no further mutual gains are
possible for the factions), then an equilibrium (which will therefore be a vv-equilibrium)
can always be interpreted as a pair of strategies in which each organization’s factions are
bargaining à la Nash, with specified bargaining powers.   Thus,  the Nash bargaining
model is ‘all we need’ to characterize any kind of efficient bargaining in a situation of
competing organizations.
4.  PUNE
In Roemer (1998, 1999) I proposed a political equilibrium concept, in which
parties with factions compete for voters.   Within parties, factions with different interests10
bargain with each other.   Having learned, six years later,  about Shapley’s paper, it is
now clear that PUNE is a special case of a vector-valued game.   In Roemer (2001), I
described  the relationship between vv-equilibrium and Nash bargaining solution
equilibrium for the political games.
The reader is referred to the above citations for a precise definition of PUNE.
The context is one in which there are two (say) political parties, competing on a multi-
dimensional policy space, which is the strategy space.   Voters are defined by their
preferences over policies.   Each party contains an ideological faction, which has a
stipulated preference order over policies, and an opportunist faction, which desires to
maximize its vote share (or the probability of victory, in another variant).   A party-
unanimity Nash equilibrium (PUNE) is a pair of policies, one played by each party, such
that neither party, facing the other’s strategy, can find a policy that makes one of its
factions better off, while not reducing the welfare of the other faction.   This is exactly a
vv-equilibrium.    (In a more articulated model, the ideological faction does not have an
exogenous preference order over policies; rather, it represents those who vote for the
party.   PUNE with ‘endogenous’ ideological factions can still be viewed as a special case
of vv equilibrium. )
An important observation about this application is that the payoff functions of the
opportunist factions are (virtually) never quasi-concave in the case of multi-dimensional
policy spaces.  Hence, the existence theorem 1 does not apply.    Indeed, I have as yet no
interesting, general existence theorem for PUNE.   However,  I find in many applications
that PUNEs exist, and they can be interpreted as Nash bargaining solution equilibria;  the
set of ordered pairs (α,β) which support Nash bargaining solution equilibria is never the11
entire unit square, but is typically a two dimensional manifold in the unit square.  Thus,
there are only certain pairs of relative bargaining powers which will support vv
equilibrium in the set-up of PUNE.
I suggest that vv equilibrium can be fruitfully applied in many areas of social
science.  One application that immediately comes to mind is oligopolistic competition,
where firms have different factions – shareholders, perhaps, who wish to maximize
profits, and managers, who wish to maximize firm size (perhaps).  If strategies are multi-
dimensional (price, quality,…) the experience with PUNE suggests that vv equilibria will
exist.    Another application could be to a simple (profit maximizing) firm competing
with a labor union that has factions with different goals (say, young and old workers, or
skilled an unskilled workers).  The strategy space for these contests, in reality, is typically
multi-dimensional: again,  one can expect vv equilibria to exist, where Nash equilibria in
pure strategies of organizations that are more simply conceived (ones with single payoff
functions) will typically not exist.
In political theory, PUNE was a solution to the non-existence of Nash equilibrium
in political games with multi-dimensional policy spaces when political actors (parties or
candidates) were conceived of as maximizing single payoff functions. Generally
speaking, in games with multi-dimensional strategy spaces, existence of Nash
equilibrium in pure strategies is often a problem: the resolution often has been to consider
mixed strategies, or to model the problem as a stage game.    I am suggesting that a third
alternative is to conceive of players as having multiple interests, and using vv
equilibrium.    And non-quasi-concavity of factional payoff functions, which is a problem
when it comes to proving general existence theorems, is in fact a good thing – because it12
reduces the size of the equilibrium manifold.   In the PUNE case, the equilibrium
manifold is often quite localized in the space S × S,  and so multiplicity of equilibrium
does not greatly reduce the predictive power of the equilibrium concept.
References
Roemer, J.E. 2001. Political Competition, Harvard University Press
Shapley, L.S. 1959. “Equilibrium points in games with vector payoffs,” Naval
Research Logistics Quarterly 6, 57-61

Political Competition,

Games with vector-valued payoffs and their application to competition between organizations

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Games with vector-valued payoffs and their application to competition between organizations

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