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the set of ultimate outcomes in Willson 1998 fails Pareto-optimality, as is seen in game 50; it fails strong dominance in games 33, 34, 36, 37; and it fails the floor axiom in
games 22, 33–37. This paper greatly generalizes and extends Willson 1998. Whereas that paper dealt
only with games with two players, the results in this paper apply to any finite number of players. The current rules of play yield proofs of more fundamental properties than in
the previous paper. This is in part true because the current system allows a richer collection of possible moves, thereby permitting a richer choice of strategies by the
players. Moreover, the new rules are arguably more realistic because they require that all players must agree before the game is over.
The rules of play given in part b fit within the tradition of the ‘Theory of Moves’ for which an extensive overview is presented in Brams 1994. Other prior analyses include
related equilibria of Marschak and Selten 1978, the limited move equilibria of Zagare 1984, the equilibria for far-sighted players of Kilgour 1984, the far-sighted equilibria
of Aaftink 1989, and the graph model for conflict resolution of Fang et al. 1993.
The ‘rules’ may also be interpreted as an arbitration procedure. The literature on arbitration is large. Some prominent studies include offer–counteroffer procedures
˚ Stahl, 1972; Rubinstein, 1982; final-offer arbitration Stevens, 1966; Wittman, 1986;
combined arbitration Brams and Merrill, 1986; sequential arbitration Brams et al., 1991. Some overviews may be found in Young 1991. Most of these studies, however,
deal with a different scenario; the system is modeled by a single numerical quantity such as the wage scale in a labor-management conflict rather than by a payoff matrix;
and most deal only with the case of two players.
2. Axioms for XG
In this section we collect some of the desirable properties possessed by the sets XG. These properties form axioms for the sets XG. The intent is that XG represents the set
of payoff vectors that might arise if rational players are playing the game exactly once. We thus seek axioms that eliminate payoff vectors which are implausible or for which
players possess strategies rendering them impossible. At this stage it is not obvious that all the properties can be met.
Formally, a game G has n players, which will be denoted 1, 2, . . . , n. For each player i, there is a set S of pure strategies. The set of all states is S 5
P
S ; thus each state is an
i i
n-tuple s 5 s , s , . . . , s in which for each i we have s [ S is the pure strategy
1 2
n i
i
selected by player i. As a result of the selection of a strategy s by each player i the state
i
s is determined, and then each player receives a profit or payoff. The payoff function p:
n
S →
R assigns to each state s the vector ps whose ith component ps 5 p s is the
i i
payoff received by player i. Each player prefers a higher payoff over a lower payoff. In some axioms, comparisons are needed between payoff vectors. We will write
ps pv to indicate that player j rates the payoff vector pv at least as well as the
j
payoff vector ps; when this happens it follows that player j’s profits satisfy ps
j
pv as well. We write ps pv if ps
pv for all players j.
j all
j
S .J. Willson Mathematical Social Sciences 39 2000 323 –348
327
1. Form axiom. Each element in XG has the form ps for some s in S. This axiom restates that only payoff vectors explicitly given by G can be obtained; no
other payoffs or side-payments are allowed. 2. Nontriviality axiom. XG is nonempty.
This axiom reflects the intuition that for any given game, there has to be some outcome. Every game must be resolved in some way. Since XG is to represent all possible
rational outcomes, XG must contain some vector.
3. Pareto-optimality axiom. If ps is in XG, there does not exist any state s9 in S such that for all i we have ps 5 ps9 and in addition, for some i, the inequality is
i i
strict. The property reflects the hope that a good negotiation process will lead to final outcomes
that are efficient. Note that it is possible that certain mixed states of outcomes which do not satisfy the form axiom would be preferred by all players over ps in XG. Thus,
perhaps, there might still exist s9 and s0 in S such that for all i, ps , 1 2
i
ps9 1 1 2 ps0 . Such a mixed strategy could not correspond to the outcome of a
i i
single play of the game. 4. Agreement axiom. If there exists a single state s such that, simultaneously for all
players i, ps is maximal, then XG contains only the vector ps.
i
Thus, in the event of complete agreement as to which strategies lead to the best outcome, the axiom states that these strategies should be selected.
5. Symmetry axiom. The sets XG should not be biased in any way by the order in which the strategies are listed nor by the order in which the players are listed. This
axiom has two aspects: a XG should depend only on the game G, hence on the sets S and the payoff function
i
p: S →
R. In particular, the order of description of the strategies in each S should be
i
irrelevant. b Suppose that s is a permutation of the n players. We think of s as merely giving a
new name for each player in the game G. We may obtain a new game G9 in which the ith player is the same as the player si in the game G, playing exactly as in the game
G. The sets T of strategy for G9 satisfy T 5 S
. Let T 5
P
T . The payoff function p9:
i i
s i i
T →
R satisfies p9s , s
, . . . , s 5 p s , s , . . . , s for each i since strategy
s 1 s 2
s n s i 1
2 n i
s played by player i in the game G9 is the same as the strategy played by player si
s i
in the game G. We require that the outcomes XG9 should correspond to the outcomes XG in exactly the same manner. Thus x , x , . . . , x lies in XG if and only if x
,
1 2
n s 1
x , . . . , x
lies in XG9.
s 2 s n
6. Lower bounds. A lower bound for player i is a number L for which there exists a pure strategy a [ S such that for all s in S for which s 5 a it is true that p s L.
i i
i
Lower bound axiom. If L is a lower bound for player i, then for all ps in XG it is also true that p s L.
i
This axiom reflects the intuition that, if player i can guarantee himself or herself a profit
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of at least L by choosing some strategy a regardless of what strategies the other players choose, then all reasonable payoffs in XG also give player i at least L. A player i can
unilaterally prevent any outcome that does not provide a profit of at least L. It does not follow that only such strategies a will arise in the settlement. Maybe there will be
agreement on another choice of strategies and player i will get more than L by playing a strategy other than a.
7. Strong dominance . The pure strategy s of player 1 is strongly dominated by the
1
9 9
9 9
strategy s if for all strategies s and s in S , s and s in S . . . , s and s in S we
1 2
2 2
3 3
3 n
n n
9 9
9 9
have ps , s , s , . . . ,s ps , s , s . . . ,s . This says that player 1 prefers the
1 2
3 n
1 1
2 3
n
9
outcome from using strategy s over the outcome from using strategy s , no matter what
1 1
the other players do, even if they change their strategies. Alternatively, no outcome when player 1 uses s is preferred by player 1 over any outcome if player 1 uses strategy
1
9
s instead.
1
It is natural to conjecture that in this circumstance player 1 will never need to play
9
strategy s because strategy s
is preferable, even if the other players change their
1 1
strategies. A ‘threat’ by player 1 to play strategy s would not be credible. We therefore
1
expect that the strategy s may be completely ignored in the computation of the
1
outcomes; we expect that there is no loss in replacing the game G by a game G9 in which strategy s does not exist. The notion generalizes in the obvious manner to each
1
player j in the game. More formally, we assume the following. Strong dominance axiom. Suppose that a player say player j has a strategy s that is
j
strongly dominated. Let G9 denote the matrix game obtained by eliminating the strategy
9 9
9
s from S . Equivalently, let S9 5
P
S , where S 5 S for i ± j, and S 5 S 2 hs j; define
j j
i i
i j
j j
p9: S9 →
R by p9 5 p u S9, and let G9 5 hS9, p9j. Then XG 5 XG9.
Note that our notion of strong dominance is not the same as the property commonly called ‘dominance’ or ‘domination’ in which the inequality above is replaced by the
9
inequality ps , s , s . . . ,s ps , s , s , . . . ,s for all s , s , . . . , s . The
1 2
3 n
1 1
2 3
n 2
3 n
difference is that in ordinary dominance the other players are not allowed to change their strategies. We definitely do not assume that we may ignore strategies that are merely
dominated, as opposed to strategies that are strongly dominated. For example, Prisoners’ Dilemma analyzed in detail in Section 7 has as its solution the outcome ps when both
players cooperate; here s contains a dominated but not strongly dominated strategy. Effectively, the strong dominance axiom lets us ignore strategies unless they can be seen
as a plausible basis for some compromise settlement acceptable to all parties.
8. Floors. The floor axiom seeks to model the default outcomes in case negotiations break down. If negotiations break down, a reasonable model is to assume that in some
order s the players will announce their irrevocable and final strategies. This order will be called the default permutation. If the players are in addition rational, then the
resulting payoff vector will be determined merely by the order s ; we will call it the ‘floor’ and denote it Fs.
More formally, let s be a permutation of h1,2, . . . ,nj. The floor Fs is the payoff
vector ps that arises from the state s 5 s , s , . . . , s obtained as follows:
1 2
n
S .J. Willson Mathematical Social Sciences 39 2000 323 –348
329
1 Player s1 announces a choice of pure strategy s [ S
.
s 1 s 1
2 Knowing that player s1 has irrevocably selected s , player s2 announces a
s 1
choice of pure strategy s [ S
.
s 2 s 2
3 Knowing the choices of s and s
, player s3 announces a choice of pure
s 1 s 2
strategy s [ S
.
s 3 s 3
. . .k Knowing the choices of s , s
, . . . , s , player sk announces a choice
s 1 s 2
s k 21
of pure strategy s [ S
.
s k s k
. . .n Knowing s , s
, . . . , s , player sn announces a choice of pure strategy
s 1 s 2
s n 21
s [ S
.
s n s n
We assume that all players are ‘rational’. Thus s1 makes a choice of s that will
s 1
maximize the resulting payoff to player s1 under the assumption that each of the other players is rational. In general player si makes a choice of the strategy s
that will
s i
maximize the resulting payoff to si , using knowledge of the previous choices of s ,
s 1
s , . . . , s
and assuming that all the other players are rational. The floor Fs is
s 2 s i 21
the outcome ps that results from all these choices. Note that saying that each player si makes the choice s
from S means that each player j makes the choice s from
s i s i
j
S . The choices determine a state s 5 s , . . . , s . Define Fs 5 ps , s , . . . , s . A
j 1
n 1
2 n
more explicit calculation of Fs is given in Theorem 6.5. We call Fs the ‘floor’ since, given s, Fs tells the lowest result in the absence of
any negotiation, assuming that all the players are rational. Any improvement in their payoffs over Fs will be on the basis of negotiation or by use of a different
permutation. The vectors Fs for different choices of s represent the fall-back or default payoffs in the event of breakdown of negotiations. The particular default
permutation s which arises, however, is unclear. Maybe a player who feels threatened would initiate the breakdown, becoming s1; or maybe some other player in
anticipation of this would instead initiate the breakdown. A player j, however, has an incentive to break down the negotiations if he would benefit from Fs for all possible
s. If a proposed outcome pv does not represent an improvement over Fs for some player i, then player i can refuse further negotiation thereby forcing the outcome to be
Fs for some permutation s rather than pv. It is therefore natural to assume that each member pv of XG is at least as good as some Fs.
Conversely, any method of breaking down negotiations and therefore leading to an outcome Fs for some permutation s provides a temptation to end negotiations leading
to the outcome Fs. Since XG must include all possible rational outcomes, it follows that XG must contain a possible outcome pv at least as good as Fs.
We therefore will assume that, if the default permutation is s, then there will be a uniquely determined outcome pv that arises at least as good for all players as Fs. Let
Xs denote this outcome pv. Floor axiom. For each permutation s of the players there is a uniquely determined
outcome Xs in XG such that Xs Fs. Moreover, every element of XG arises
all
in this manner for some permutation s. According to the floor axiom, XG consists of all the payoff vectors Xs. It is quite
possible that different default permutations s give rise to the same payoff vector Xs; hence it is still possible, for example, that XG is a singleton set.
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9. Reduction using the default outcomes. Suppose that s is a permutation of the players
h1,2, . . . ,nj. If the negotiations break down and players make final irrevocable choices of strategy in the order s, then the
outcome will be the floor Fs. For definiteness, let Xs ; G denote the Xs that arises in game G using the permutation s, and similarly Fs ; G denotes the floor arising from
game G using s. Thus Xs ; G is a payoff vector pv for some state v, and Xs ; G
Fs ; G.
all
Our intuition suggests that, given this s as the default order of play if negotiations fail, the only relevant payoff vectors pv satisfy that pv
Fs. No other pv
all
would be relevant since some player would instead refuse negotiation and instead the game would end up at Fs. It is therefore natural to define for each permutation s a
modified game Gs as follows in which the irrelevant payoff vectors are replaced by Fs.
More explicitly, suppose G is the game with strategies S and payoff vectors p:
n n
S →
R . Let Gs be the game with strategies S and payoff vectors q: S →
R given by qs 5 ps if ps
Fs ; G;
all
qs 5 Fs ; G otherwise. We call Gs the game G reduced by the permutation s because the possible payoffs
for the game have been reduced using the default permutation s. Since only the payoffs in Gs are relevant to the game when the default permutation is
s, the players may as well utilize Gs rather than G in determining Xs; G if s is known. Hence Xs ; G may be determined from the game Gs, and the outcome in G
given the default permutation s will be the same as that for s in the reduced game Gs. This intuition is the basis for the next axiom.
Default reduction axiom. For each permutation s, Xs; G 5 Xs; Gs.
Remark. The axioms stated above are not independent. Some axioms imply others. For example, the Pareto-optimality axiom implies the agreement axiom, and the floor axiom
implies the lower bound axiom. In this section such redundant properties are stated because the strongest axioms such as the floor axiom could be regarded as ques-
tionable; the reader might seek to reject some axioms and seek some weaker versions.
Following are several examples of matrix games with analyses using the axioms described above. The games have two players. Player 1 may be called Row or R and
selects the row of the matrix; player 2 is called Column or C and selects the column of the matrix. The payoff vector from the ith row and the jth column has form a, b, where
a is the profit to player 1 and b is the profit to player 2. Larger numbers are better.
Example 1. Use of the lower bound and Pareto-optimality axioms.
Consider the matrix 3, 4
1, 1 2, 3
4, 2
S .J. Willson Mathematical Social Sciences 39 2000 323 –348
331
Each X in XG must be Pareto-optimal, so the only possibilities are 3, 4 and 4, 2. The latter violates the lower bound axiom since Column can guarantee herself at least 3
by selecting the first column. This leaves 3, 4 as the only possibility. Since XG is nonempty it follows that XG5
h3, 4j.
Example 2. Use of the floor axiom.
2, 4 3, 3
1, 2 4, 1
Row can guarantee herself at least 2 by selecting the first row, and Column can guarantee herself at least 2 by selecting the first column. Hence the lower bound axiom
eliminates 1, 2 and 4, 1. We next show that FCR 5 2, 4. If Column initiated a breakdown of negotiations
and selected the first column irrevocably, then the result would be 2, 4 since Row would select the first row to get a better payoff. If, instead, Column selected the second
column irrevocably, then Row would select the second row to obtain the better payoff 4, 1. Column prefers 2, 4 over 4, 1 and hence would select the first column when
initiating a breakdown. Therefore FCR 5 2, 4. Similarly we can compute FRC 5 2, 4.
All breakdowns of the negotiations therefore lead to the payoff vector 2, 4, and the floor axiom eliminates 3, 3 as a possible outcome. Effectively, Column could benefit
from a breakdown of negotiations unless 2, 4 were being considered. Hence XG5 h2, 4j.
Example 3. Use of the strong dominance axiom.
3, 4 4, 3
1, 2 2, 1
The strong dominance axiom says that the outcomes should be the same as for the matrix game G9 given by
3, 4 4, 3 since the last row of G is strongly dominated by the first. For G9, Column can guarantee
herself at least 4 by choosing the first column, so the lower bound axiom then implies that the only possible outcome is 3, 4. Hence XG5
h3, 4j.
Example 4. Use of the default reduction axiom.
1, 4 7, 1
4, 6 9, 2
2, 5 8, 3
3, 9 5, 8
6, 7 The lower bound axiom says that Row can guarantee herself at least 3 and Column can
guarantee himself at least 3. The Pareto optimal outcomes that are consistent with this are precisely 8, 3, 3, 9, 5, 8, 6, 7. Since FRC 5 4, 6 while FCR 5 8, 3, the
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floor axiom eliminates 3, 9 as a possibility. Since 8, 3 is Pareto-optimal and XCR
FCR, it follows that XCR 5 8, 3. The floor axiom, however, does not
all
determine XRC . There are two choices of Pareto-optimal payoffs that each player prefers over FRC 5 4, 6, namely 5, 8 and 6, 7. The floor axiom is thus consistent
with either XRC 5 5, 8 or XRC 5 6, 7. To decide among these, we utilize the default reduction axiom. Since FRC 5 4, 6,
the reduced game GRC is obtained by replacing payoffs not 4, 6 by 4, 6. It
all
therefore has payoff matrix 4, 6
4, 6 4, 6
4, 6 4, 6
4, 6 4, 6
5, 8 6, 7
In this matrix, the last row strongly dominates the first two rows, so the strong dominance axiom reduces the analysis to the matrix
4, 6 5, 8 6, 7 and the lower bound axiom then leads to outcome 5, 8. Hence XRC; GRC 5 5, 8,
whence, by the default reduction axiom, XRC; G 5 5, 8. It follows that XG 5
hXRC, XCRj 5 h5, 8, 8, 3j.
3. Rules of play
