Rationalization of Collective Choice Functions by Games with Perfect Information
by
Yongsheng Xu
Department of Economics, Andrew Young School of Policy Studies
Georgia State University, Atlanta, GA 30303
and
Lin Zhou
Department of Economics, WP Carey School of Business
Arizona State University, Tempe, AZ 85287
First Version
January 30, 2004
Abstract: Collective choices are often cyclic and cannot be rationalized
by a well-defined linear ordering. In this paper we identify conditions
under which collective choices, potentially cyclic, can be rationalized by
games with perfect information.
JEL Classification Numbers: C70, D70
-1-
1.
Introduction
In this paper we study collective decision- making in a choice-based theoretic model.
Suppose X is a (finite) set of all potential alternatives and C is a choice function on X that
chooses a unique alternative C(A) for every nonempty subset A of X. The interpretation of
C(A) is that it is the final alternative chosen by a group of individuals when the set of
available alternatives is A. The choice function C is a (highly) reduced form of collective
choices that contains all information we, as outside observers, have. We do not know
who these individuals are, let alone the specific decision-making process they might use
in reaching their choices.
We need to say a few words regarding the relevance of our model before we proceed.
When a collective body is a complete democracy, we know everything about its
membership and rules under which it operates. In this case, our model is not applicable.
However, there are many other cases where decisions have been made behind the doors:
for example, the political bureau of the former Soviet Communist Party, or the board of
directors of Tyco under Mr. Kozlowski’s reign. Even though the nominal members of
these two examples were known, it was unclear who really had powers in decisionmaking and how they exercised their powers. The model in this paper deals with these
types of situations. (Two recent papers by Sprumont (2000) and Ray and Zhou (2001)
have studied collective choices under well-defined membership and decision-making
procedures.)
There are two tasks ahead of us: the first is to define in what sense a choice function
C is considered collectively rationalizable; and the second is to find conditions under
which a choice function C is indeed collectively rationalizable.
To motivate our definition of collective rationalizability, we start with a brief
discussion of individual ratio nalizability. When C is considered an individual choice
function, the rationalizability of C is often identified with the representability of C by a
linear ordering R. Formally, a choice function C can be individually rationalized by a
linear ordering R if and only if
-2-
(IR)
C(A) = B(A; R) for all A ⊆ X with A ≠ ∅ ,
in which B(A; R) is the best element of R on A. Conditions for a choice function to be
individually rationalizable have been developed by many authors, including Houthakka
(1950), Arrow (1959), Richter (1966), and Sen (1971), etc. In particular, an individually
rationalizable choice function C must be acyclic, i.e., there should be no triple x, y, and z
that form a cycle with
(CYC)
C({x, y}) = x, C({y, z}) = y, and C({z, x}) = z .
However, collective choices often involve cycles. For example, consider a simple
decision rule with three alternatives X = {x, y, z} and two agents. For any set of
alternatives A ⊆ X, agent 1 has the option to pick x first whenever x ∈ A. If agent 1 does
not pick x, the n x is eliminated and agent 2 can pick any alternative from the remaining
set. This rule can be represented by the following tree:
Moreover, suppose agent 1’s linear ordering is: z R1 x R1 y, and agent 2’s linear ordering
is: y R2 x R2 z. When both agents act rationally in the sense that they always play
subgame perfect Nash equilibria, this decision rule, under the designated preferences,
generates outcomes that are consistent with (CYC). (In addition to choices specified in
(CYC), x is the outcome when all alternatives are available.) As a result, this game tree
rationalizes a collective choice function with cycles.
This example leads us to adopt the following notion of the collective rationalizability
of a choice function C. Suppose C is a choice function defined for a finite set X. First, we
construct a tree G that has alternatives in X as terminal nodes. Second, we designate
-3-
linear orderings R’s for all agents in this game. We say that the choice function C is
collectively rationalized by G if
(CR)
C(A) = SPNE(G|A; R), for all A ⊆ X with A ≠ ∅,
in which G|A is the reduced game of G that is derived from G by retaining only paths that
lead to terminal nodes in A, and SPNE stands for subgame perfect Nash equilibrium. 1
Compared with (IR), (CR) allows us more freedom when we try to rationalize a
choice function. First, we can introduce more agents with each of them having a different
linear ordering; second, we can construct game trees that have much richer structures
than the plain individual utility maximization process. Yet we are not totally free with
(CR). For example, any choice function that satisfies (CR) must respect unanimity, i.e.,
C({x, y}) = x for all y∈ A ⇒ C({x}∪A) = x.
While there are many other conditions we can go through, the challenge we face is to
find conditions that are both necessary and sufficient for a choice function to be
rationalized by a game tree. The main contribution of this paper is to identify a pair of
conditions, which together characterize choice functions that can be rationalized by game
trees. The first condition -- the weak separability -- is a type of path independence
condition (see Plott (1973)) and the second condition – the divergence consistency -deals with choices when cycles are intertwined with each other. In Section 2 we provide
the formal definitions of these conditions and discuss why they are necessary for a choice
function to be rationalized by a game tree. In Section 3 we prove the main result of the
paper that these two conditions are sufficient for a choice function to be rationalized by a
game tree. We conclude the paper with more discussions and remarks in Section 4.
1
Note that a reduced game G|A of G is not necessarily a subgame of G: It is a subgame of G only when A
consists of terminal nodes for a particular subgame of G.
-4-
2.
The Basic Set-Up
The (finite) set of alternatives is X = {x, y, z, …}. A choice function C is a mapping
from 2X\∅, the set of all non-empty subsets of X, to X with C(A) ∈ A for all A ∈ 2X \∅.
In this paper we use extensive games of perfect information, or game trees for short,
to rationalize choice functions. For a given choice function C defined on X, we consider
any game tree G that has a one-to-one mapping from all its terminal nodes to X. Without
loss of generality, we shall identify G’s terminal nodes with X. For any A ∈ 2X \∅, G|A is
the reduced game tree of G that retains only branches that lead to terminal nodes in A. For
example, let X = {x, y, z} and G is the game tree we described in the introduction:
then, for A1 = {x, y} and A2 = {y, z}, the reduced game trees G|A1 and G|A2 are:
G|A1
G|A2
We say that a choice function C is rationalized by a game tree G if there is a game
tree G that has all alternatives in X as terminal nodes and
(CR)
C(A) = SPNE(G|A) , for all A ∈ 2X\∅,
in which SPNE(G|A) stands for the subgame perfect Nash equilibrium of the reduced
game G|A.
-5-
If a choice function C is rationalized by some game tree G, we also say that C is
collectively rationalizable.
There is a considerable degree of freedom in constructing a game tree to rationalize a
choice function C: we are free to choose the number of players, the structure of the tree,
and the preference relation of each player on X. However, there are so many possible
choice functions that only a small number of choice functions are collectively
rationalizable. To see it from a different angle, let us investigate some restrictions that
collective rationalizability imposes on choice functions.
First, when a choice function C is rationalized by a game G of perfect information,
the choice function must be in some sense weakly separable. When we split the game tree
G at the root by separating any particular initial branch from the rest of the tree, we also
partition X into two non- degenerate disjoint sets Y and Z. The fact that C is collectively
rationalizable by G implies that C(X) = C(Y∪Z) = C({C(Y), C(Z)}). Moreover, this
property should hold for all subsets of X, Y, and Z. Hence, any choice function C that is
collectively rationalizable must satisfy:
Weak Separability. For any A ∈ 2X \∅ with | A| > 1, there exist a non-degenerate
partition A = B∪D (B∩D = ∅, B / ∅, and D / ∅) such that
(WS) C(S∪T) = C({C(S), C(T)}), for all S ⊆ B and T ⊆ D with S / ∅, and T / ∅.
If we strengthen WS by requiring that (WS) should hold for all partitions of A, we obtain
strong separability, which is both necessary and sufficient for rationalizability of a choice
function by a single linear ordering on X. The weak separability is also associated with
the single preference rationalization in another way as the following proposition shows.
Proposition. If a choice function C satisfies weak separability and is acyclic, then C can
be rationalized by a single linear ordering on X, and vice versa.
The proof of the proposition is straightforward. First, we define a binary relation RC on X
by: x RC y iff C({x, y}) = x. Obviously, RC is complete, reflexive, and for all x, y ∈ X: [(x
-6-
RC y and y RC x) ⇒ x = y]. Since C is acyclic, RC is also transitive. Therefore, RC is a
linear ordering. We then use weak separability to show that C(A) = B(A; RC) for all A ⊆
X (by induction on the number of alternatives contained in A).
It is clear we need to drop acyclicity in our inquiry. The question is what condition
can replace acyclicity, which together with weak separability will enable us to obtain
collective rationalizability. To answer this question, let us study cycles induced by a
choice function, in particular, cycles that consist of three alternatives.
For any triple x, y, and z, we say x, y, and z form a 3-cycle if C({x, y}) = x, C({y, z}) =
y, and C({z, x}) = z, or if similar conditions hold for a permutation of x, y, and z. (We
may study cycles that contain more alternatives. However, if a choice function induces a
cycle of any number of alternatives, it must induce a cycle of three alternatives. This
explains why we choose to focus on 3-cycles.)
When x, y, and z form a 3-cycle, choices over {x, y, z} and its subsets can be
represented by a game tree that depends on C({x, y ,z}). For example, when C({x, y}) = x,
C({y, z}) = y, and C({z, x}) = z, and C({x, y ,z}) = x, the choice function can be
represented by the following tree:
1
2
z
x
y
y
z
Other than re- labeling of players or terminal nodes, this game tree is (almost) unique.
The player who chooses first can opt for x, or pass x and let the other player choose
between y and z. Also the first player must rank x between y and z, and the second
player’s ranking of y and z must be the opposite of that of the first player.
-7-
Since we shall encounter 3-cycles repeatedly, we adopt a convenient terminology: For
any triple x, y, z, we say that x diverges before y and z, if x, y, and z form a 3-cycle and
C({x, y, z}) = x.
When there are more than three alternatives, a choice function may ha ve several 3cycles. Since each 3-cycle uniquely determines the structure of a branch of any potential
game tree that represents the choice function, these 3-cycles must overlap properly for a
choice function to be rationalized by a game tree.
Consider a situation in which X = {x 1 , x2 , y1 , y2 }. If a choice function C is such that x 1
diverges before y1 and y2 , and y1 diverges before x 1 and x 2 , then for C to be rationalizable
by a game tree, the branches of the game must look like the following.
In addition, player 1 must rank x 1 between y1 and y2 , and rank y1 between x 1 and x 2 . Now
there are two possible cases:
First, x 1 is ranked above y1 , or more precisely, C({x 1 , y1 }) = x1 . Then C({x 1 , y2 }) = y2
(since x 1 is between y1 and y2 ) and C({x 2 , y1}) = y1 (since y1 is between x 1 and x 2 ). In this
case, player 1’s linear ordering R1 must be: y2 R1 x1 R1 y1 R1 x2 . Hence, C({x 2 , y2}) = y2 .
Second, x 1 is ranked below y1 , or C({x 1 , y1 }) = y1 . Then C({x 1 , y2 }) = x1 (since x 1 is
between y1 and y2 ) and C({x 2 , y1 }) = x2 (since y1 is between x 1 and x 2 ). In this case, player
1’s linear ordering R2 must be: x 2 R2 y1 R2 x1 R2 y2 . Hence, C({x 2 , y2 }) = x 2 .
To summarize, if a choice function C can be rationalized by a game tree, then it must
satisfy the following condition:
-8-
Divergence Consistency. For any four alternatives x 1 , x2 , y1 , y2 ∈ X, if x 1 diverges before
y1 and y2 , and y1 diverges before x 1 and x 2 , then C({x 1 , y1}) = x1 iff C({x 2 , y2 }) = y2 .
It turns out that divergence consistency and weak separability together are sufficient
for a choice function to be collectively rationalizable.
Theorem. A choice function C can be rationalized by a game tree if and only if it
satisfies weak separability and divergence consistency.
The proof is given in the next section. Before moving on, we present two examples
showing that the conditions of weak separability and divergence consistency are
independent when |X| > 3.
Example 1. Let X = {x 1 , x2 , y1 , y2 }. Partition it into two sets {x 1 , x2 } and {y1 , y2 }.
Consider a choice function C1 with
(1a) C1 ({x 1 , y1}) = x1 , C1 ({x 1 , y2 }) = y2 , C1 ({x2 , y1 }) = y1 , C({x2 , y2}) = x2 ;
(1b) C1 ({x 1 , x 2}) = x2 , C1 ({y1 , y2 }) = y1 ;
(1c) C1 ({x 1 , x 2 , y1 }) = C1 ({C1 ({x 1 , x2 }), y1}) = C1 ({x2 , y1}) = y1 ,
C1 ({x1 , x2 , y2 }) = C1 ({C1 ({x 1 , x2 }), y2}) = C1 ({x2 , y2}) = x2 ,
C1 ({x1 , y1 , y2 }) = C1 ({x 1 , C1 ({y1 , y2})}) = C1 ({x1 , y1}) = x1 ,
C1 ({x2 , y1 , y2 }) = C1 ({x 2 , C1 ({y1 , y2})}) = C1 ({x2 , y1}) = y1 ;
(1d)
C1 ({x 1 , x 2 , y1 , y2 }) = C1 ({C1 ({x1 , x2 }), C1 (y1 , y2})) = C1 ({x 2 , y1 }) = y1 .
It is clear by construction that C1 satisfies weak separability. However, C1 does not
satisfy divergence consistency since x 1 diverges before y1 and y2 , and y1 diverges before
x1 and x 2 , yet C1 ({x 1 , y1 }) = x1 and C({x 2 , y2}) = x2 .
-9-
♦
Example 2. Let X = {x 1 , x2 , y1 , y2 }. Fix a linear ordering R on X with x 1 Rx 2 R y1 Ry2 .
Consider a choice function C2 that maximizes R on any subset of X with three alternatives
or less but C2 (X) = y2 . It is clear that C2 does not satisfy weak separability for A = X since
y2 can never be chosen in a pairwise comparison. On the other hand, C2 trivially satisfies
divergence consistency since C2 has no 3-cycles.
3.
♦
The Proof of the Main Result
We have already demonstrated the necessity of these two conditions in Section 2. We
now show that these two conditions together are sufficient for a choice function to be
rationalizable by a game tree. We prove this by induction on the number of the
alternatives X contains.
The result is trivial for |X| = 2. Now assume that for some n, any choice function that
satisfies weak separability and divergence consistency on a set X with |X| ≤ n can be
rationalized by a game tree with no more than |X| -1 players. Consider choice functions
defined on some set X with |X| = n + 1. When a choice function C satisfying weak
separability and divergence consistency on X, we can find a non-degenerate partition X1
and X2 of X such that
C(S∪T) = C({C(S), C(T)})
for all S ⊆ X1 and T ⊆ X2 .
By the induction hypothesis, there are two game trees, G1 for X1 and G2 for X2 , such that
C is rationalized by G1 on X1 and by G2 on X2 . Then, construct the following game tree G:
The number of players needed in G1 and G2 is no more than |X| -1. Here we just let
player 1 be the one who is not in G1 or G2 . Notice that preferences on X2 by players in G1
- 10 -
are immaterial for rationalization of C, and vice versa. So, as long as we can construct a
linear ordering for player 1 that is consistent with C, then we are done. There are three
possibilities: (a) |X1 | = 1; (b) |X2 | = 1; and (c) |X1 | >1 and |X2 | > 1.
(a) |X1 | = 1, say X1 = {x}. We define player 1’s preference relation as follows. For all
y ∈ X2 , xR1 y if C(x, y) = x, and yR1 x if C(x, y) = y. Clearly, this preference relation
is incomplete. However, it has no cycles. Hence, it can be extended to a linear
ordering R1* on X.
For any set A ⊆ X with x∈ A, A = {x}∪B with B ⊆ X2 . Then,
C(A) = C({x}∪B)
= C({x, C(B)})
(by weak separability)
= C({x, SPNE(G2 |B)})
(by induction hypothesis)
= SPNE(G|A) .
Similarly, for any set A ⊆ X with x∉ A,
C(A) = SPNE(G2 |A))
(by induction hypothesis)
= SPNE(G|A) .
(b) |X2 | = 1. This case can be dealt with similarly as case (a) above.
(c) |X1 | >1 and |X2 | > 1. Define player 1’s preference relation R1 for pairs with one
alternative in X1 and another in X2 as follows: for all x ∈ X1 , all y ∈ X2 , xR1 y if
C({x, y}) = x, and yR1 x if C({x, y})= y. For pairs with both alternatives in X1 or
both in X2 , R1 is not yet defined.
We now show that R1 does not have cycles. Suppose to the contrary that R1
has a cycle. Given the nature of R1 , there must exist distinct x 1 , … , x k ∈ X1 , and
distinct y1 , …, yk ∈ X2 such that x 1 R1 y1 , y1 R1x2 , x2 R1 y2 , … , xkR1 yk, ykR1x1 .
Moreover, we may assume there is a cycle with k = 2. (If y2 R1 x1 , then this is it. If
not, then x 1 R1 y2 and we can drop x 2 and y1 from the cycle and reduce its length.
- 11 -
This is repeated until a cycle with k = 2 is found.) Hence, we have x 1 , x2 ∈ X1 , and
y1 , y2 ∈ X2 such that x 1 R1 y1 , y1 R1x 2 , x2 R1 y2 , y2 R1x 1 , or
(0)
C({x 1 , y1}) = x1 , C({y1 , x 2}) = y1 , C({x2 , y2}) = x2 , C({y2 , x 1}) = y2 .
There are four possible subcases concerning C({x 1 , x 2}) and C({y1 , y2 }):
(i)
C({x 1 , x 2}) = x1 and C({y1 , y2 }) = y1 ;
(ii)
C({x 1 , x 2}) = x1 and C({y1 , y2 }) = y2 ;
(iii) C({x 1 , x 2}) = x2 and C({y1 , y2 }) = y1 ;
(iv) C({x 1 , x 2}) = x2 and C({y1 , y2 }) = y2 .
Consider (i). Together with (0), C({x 1 , x2 }) = x1 implies x 1 , x2 , y2 are a 3-cycle.
Also, C({x 1 , x 2 , y2 }) = C({C({x 1 , x 2}), y2 }) = C({x1 , y2}) = y2 . Hence y2 diverges
before x 1 and x 2 . Together with (0), C({y1 , y2 }) = y1 implies x 1 , y1 , y2 are a 3-cycle.
Also, C({x 1 , y1 , y2 }) = C({x1 , C({y1 , y2 })}) = C({x1 , y1}) = x2 . Hence x 1 diverges
before y1 and y2 . Then, by divergence consistency, C({x 1 , y2 }) = y2 should lead to
C({x 2 , y1}) = x2 . But this contradicts (0).
Next consider (ii). Again C({x 1 , x 2}) = x1 and (0) imply y2 diverges before x 1 and
x2 . Together with (0), C({y1 , y2 }) = y2 implies x 2 , y1 , y2 are a 3-cycle. Also,
C({x 2 , y1 , y2 }) = C({x2 , C({y1 , y2})}) = C({x 2 , y2 }) = x2 . Hence x 2 diverges before
y1 and y2 . Then, by divergence consistency, C({x 2 , y2 }) = x2 should lead to
C({x 1 , y1}) = y1 . But this again contradicts (0).
We can repeat the same argument for (iii) and (iv) and demonstrate contradictions
there. Therefore, R1 cannot have cycles.
Since R1 has no cycles, from the definition of R1 , it can be extended to a
preference relation that is linear on the entire X. Finally, we can use an argument
similar to that in (a) to show that the game tree G rationalizes C on X. This
completes the induction.
- 12 -
4.
Some Remarks
In this paper we have derived conditions that are necessary and sufficient for choice
functions to be rationalized by extensive games with perfect information. There are
several possible extensions of the main result.
We have assumed that a choice function C defined on X is to be rationalized by such
a game tree G that each alternative x ∈ X appears as a terminal node of G once and once
only. We can modify this assumption and allow each alternative to appear possibly
multiple times. Of course, this modification calls for a reinterpretation of the reduced
game G|A. One interpretation is that G|A retains all branches of G that lead to alternatives
in A. How much will this modification change our result? Does this modification allow
more choice functions to be collectively rationalizable? The short answer is yes. For
example, let us consider the choice function with three alternatives that is generated by a
“voting by veto” game G. In this game, player 1 first has the option of vetoing at most
one alternative, then player 2 picks an alternative from those that remain. Assume
players’ linear orderings are as given next to the tree in the following graph.
Now we can calculate the subgame perfect Nash equilibrium outcome of G|A for each
A ⊆ X with A ≠ ∅. This generates a choice function C with:
C({x, y}) = x , C({x, z}) = x, C({y, z}) = z, and C({x, y, z}) = z .
It is easy to verify that C does not satisfy weak separability. In fact, with three
alternatives, any choice function can be rationalized by some game tree when we allow
- 13 -
each alternative to be terminal nodes of a tree multiple times. The interesting question is
whether this is true in general. While we would like to identify a set of conditions that are
necessary and sufficient for choice functions to be collectively rationalized, it is also
conceivable that any choice function might be rationalized in this extended setting.
There is another possible extension in a different direction. Here we will maintain the
assumption that each alternative appears on the game tree only once. However, we relax
the assumption that all choice functions are single-valued by allowing the possibility of
multi- valued choice correspondences. Now we have to allow players’ preferences to be
indifferent for certain pairs of alternatives in order to have multiple subgame perfect
Nash equilibria. What conditions are necessary and sufficient for multi- valued choice
correspondences to be collectively rationalizable? Obviously, weak separability continues
to be necessary. So does divergence consistency. Yet together they are not sufficient for a
multi- valued choice correspondence to be collectively rationalizable. Consider a choice
correspondence with three alternatives: X = {x, y, z}, C({x, y}) = {x, y}, C({x, z}) = {x, z},
C({y, z}) = {y, z}, and C({x, y, z}) = {x}. Clearly, this choice correspondence satisfies
weak separability (any partition of {x, y, z} is fine), as well as divergence consistency
(there are no cycles). However, C cannot be rationalized by any game tree. If C were
rationalized by some game tree G, then C({x, y}) = {x, y}, C({x, z}) = {x, z}, C({y, z}) =
{y, z} would imply C({x, y, z}) = {x, y, z} ≠ {x}. Hence, this choice correspondence is
not collectively rationalizable. We do not have a complete solution for this extension yet.
Finally, we can also give our model an individual decision theory interpretation.
Many researchers have reported cases in which individual choices exhibit cycles.
Particularly, such cycles are common occurrence when an individual uses different
criteria in evaluating various alternatives at different points. For example, Katie plans to
go out for dinner tomorrow. She cares about both the healthiness and the taste of the food.
The restaurant she plans to go to has three dishes are on the menu: T-bone steak, sushi,
the monk’s delight (a vegetarian special). In terms of healthiness, Katie ranks the monk’s
delight the highest, and T-bone steak the lowest. In terms of taste, however, Katie ranks
T-bone steak the highest, and the monk’s delight the lowest. If she wants sushi, she has to
order one day in advance since fresh seafood has to be pre-ordered. But she can decide
- 14 -
until she arrives at the restaurant if she wants T-bone or the monk’s delight. Hence her
decision tree is:
Katie knows that her sensible choice should be made based on the healthiness of the food.
Yet she anticipates that her preference for healthy food will succumb to her preference
for tasty food once she steps in the restaurant and sits down at the dinner table. This leads
to a situation that is exactly the same game tree in the introduction with player 1 and
player 2 there being replaced by Katie’s split personalities of tonight and tomorrow. 2 In
general, we can replace different players in a game tree by one player’s different
preferences at different decision nodes, then our result in this paper also provides
potential insight to the nature of cycles of individual choices.
2
A similar example in a slightly different context is included in Ok and Masatlioglu (2003).
- 15 -
References
Arrow, K. (1959): “Rational choice functions and orderings,” Economica, 26, 121-127.
Houthakker, H. S. (1950): “Revealed preference and utility function,” Economica, 17,
159-174.
Ok, Efe, and Y. Masatlioglu (2003): “A general theory of time preference,” Discussion
Paper, New York University.
Plott, C. (1973): “Path independence, rationality and social choice,” Econometrica, 41,
1075-1091.
Ray, I. and L. Zhou (2001): “Game theory via revealed preferences,” Games and
Economic Behavior, 37, no. 2, 415-424.
Richter, M. (1966): “Revealed preference theory,” Econometrica, 34, 635-645.
Sen, A. K. (1971): “Choice functions and revealed preference,” Review of Economic
Studies, 38, 307-317.
Sprumont, Y. (2000): “On the testable implications of collective choice theories,” Journal
of Economic Theory, 93, no. 2, 205-232.
- 16 -