332 S
.J. Willson Mathematical Social Sciences 39 2000 323 –348
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
In this section we present the ‘rules of play’ for a dynamic determination of XG. In later sections we will show that the final outcomes obtained by these rules of play in fact
satisfy the axioms of Section 2. 1. Each player i selects an initial pure strategy s [ S . Together, these strategies yield
i i
an initial state s, which becomes the first ‘current state’ of the game. 2. Randomly, a permutation s of the players is selected. Assume that each of the n
permutations has positive probability. This permutation s gives the order in which the players will make their moves for each round of the play.
3. A sufficiently large integer r is selected. This number r is the maximum number of rounds that can take place, and it is announced to all players. In fact, the arguments
in Theorem 5.4 give an easy way to estimate the minimum size t for r. As long as r 5 t, the exact choice of r will have no effect on the outcome. The players need only to have
the possibility of playing at least t rounds.
4. A ‘move’ by player i consists of a choice of a pure strategy from S . Before the
i
9
move, there is a current state s; after the move, there is a current state s9 such that s 5 s
j j
9
for all j ± i, but s is the state that player i selects during this move. It is possible that
i
9
s 5 s and hence s9 5 s if player i reasserts the previous choice; it is also possible that
i i
9
s equals any other member of S . Player i is completely free to choose any state from S
i i
i
during the move. No other player j with j ± i can change s during player i’s move.
j
5. A ‘round’ consists of n moves in the order given by the permutation s. Before the
S .J. Willson Mathematical Social Sciences 39 2000 323 –348
333
round begins, there is a current state s. When the round begins, player s1 makes a move, yielding a current state s9. When this move is completed, player s2 makes a
move, yielding a current state s0. When this move is completed, player s3 makes a move. This process continues until player sn makes a move, completing the round at a
current state v.
6. After each round, ending in a current state v, all players are asked whether they wish to stop the game. If all the players agree to stop the game, then the game ends with
payoff vector pv. Thus player i receives payoff p v, where v is the current state. If the
i
maximum number r of rounds has been played, leading to the current state v, then the game likewise ends with payoff vector pv; no more rounds are allowed. If the
maximum number r of rounds has not yet been reached, and even just one player does not agree to stop the game after a certain round, then the game continues with another
round.
It is clear that ultimately a final payoff vector pv is obtained by this process since there are at most r rounds and each consists of n moves. We will let XG denote the set
of final payoff vectors pv that will arise.
Remark. In fact, we will see in Section 6 that if rational players use these rules, then the outcome will not depend at all on the initial pure strategies selected in step 1 nor on the
number r selected in step 3, as long as r satisfies a condition given in Section 6. For example, we will see that the possibly arbitrary choice of r in step 3 could be replaced
by the following deterministic step 39:
39. Suppose that the number of pure strategies for player i is c 5 uS u. Let r 5
P
c .
i i
i
Announce to all players that there will be at most r rounds. Consequently the only randomness that may affect the final payoff vector is the choice
of the permutation s in step 2.
4. Rationality and strictness
