Running head: Potential Maximization Potential Maximization and Coalition Government Formation * Rod Garratt † , James E. Parco ‡ , Cheng-Zhong Qin † , and Amnon Rapoport # † University of California, Santa Barbara Department Economics, Santa Barbara, CA 93106 ‡ United States Air Force Academy Department of Management, Colorado Springs, CO 80840 # University of Arizona Department of Management and Policy, Tucson, AZ 85721 March 8, 2002 Original version: June 16, 2001 Corresponding author: Rod Garratt [email protected]Phone: (805) 893-2849 Fax: (805) 893-8830 Homepage: http://www.econ.ucsb.edu/~garratt/faculty/garratt.htm Grand Coalition web site: http://www.grandcoalition.com * We would like to thank Maya Rosenblatt for help in data collection. Amnon Rapoport would like to acknowledge financial support of this research by a grant from the Hong Kong Research Grants Council (Project No. CA98/99.BM01) awarded to the Hong Kong University of Science and Technology.
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Running head: Potential Maximization
Potential Maximization and Coalition GovernmentFormation*
Rod Garratt†, James E. Parco‡, Cheng-Zhong Qin†, and Amnon Rapoport#
†University of California, Santa BarbaraDepartment Economics, Santa Barbara, CA 93106
‡United States Air Force AcademyDepartment of Management, Colorado Springs, CO 80840
#University of ArizonaDepartment of Management and Policy, Tucson, AZ 85721
March 8, 2002Original version: June 16, 2001
Corresponding author:
Rod [email protected]: (805) 893-2849Fax: (805) 893-8830Homepage: http://www.econ.ucsb.edu/~garratt/faculty/garratt.htmGrand Coalition web site: http://www.grandcoalition.com
*We would like to thank Maya Rosenblatt for help in data collection. Amnon Rapoport would like to acknowledgefinancial support of this research by a grant from the Hong Kong Research Grants Council (Project No.CA98/99.BM01) awarded to the Hong Kong University of Science and Technology.
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Abstract
A theory of coalition government formation is presented in which non-minimal winning
coalitions may form in equilibrium. Theoretical predictions for five games are derived and tested
experimentally. The experimental data support potential maximization as a refinement of Nash
equilibrium. In particular, they support theoretical predictions that (1) non minimal winning
coalitions occur when the distance between policy positions of the parties is large relative to the
value of forming the government, and (2) in cases where the distance between policy positions of
the parties is small relative to the value of forming the government, a minimal winning coalition
forms that includes the two parties that are closest together on the policy line. Potential
maximization is competitively tested against other solution concepts including iterated
elimination of dominated strategies and strong/coalition proof Nash equilibrium.
This paper applies recent theoretical advances in the theory of potential games to coalition
formation, derives predictions and tests them experimentally. The goal is to ascertain to what
extent potential maximization (a refinement of Nash equilibrium) is a useful guide in predicting
coalition formation behavior. We consider three-player games. This is the simplest setting in
which interesting coalitional structures can emerge. Moreover, the theoretical predictions for the
three-player case are unambiguous and reduce to few enough cases that exhaustive experimental
testing is feasible.
The context of our study is coalition government formation. Players are political parties
that are differentiated by exogenously given policy positions.1 The game begins after the election
is over. To avoid trivial cases, it is assumed that no party controls a majority of the votes, and
hence any two parties (or all three) control a majority.2 The coalition that forms the government
is the one controlling a majority of the votes cast. The members of the government are entitled to
share a surplus, which is interpreted as the value of forming the government. The amount of the
surplus depends on the policy positions of the government members and the policy choice of the
government. Assuming distinct policy positions, adding more parties to a coalition government
lowers the value of forming the government as a greater deal of compromising is required to
accommodate a wider spectrum of beliefs. The technical implication is that the characteristic
function of the coalitional game is not superadditive.
1 Policy positions are points on the Real line. This is not essential. Assuming a one-dimensional policy space allowsus to classify predictions for all three-player games in terms of just two distances. This reduces the number of casesto test experimentally and is appropriate for the purpose at hand.2 The model of coalition government formation applies more to European countries and Canada than to the UnitedStates. Votes may be thought of as seats in a parliamentary system.
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The surplus of the coalition government is divided according to the Myerson value
(Myerson, 1977). The Myerson value is a generalization of the Shapley value (Shapley, 1953)
that reflects the cooperation (or link) structure within each coalition (see also Aumann &
Myerson, 1988). If all the members of a coalition are linked, the Myerson value is the same as
the Shapley value; otherwise, more weight is given to players that hold special positions in the
link structure.
The attachment of more weight to players with a special position in the link structure
makes sense in this context if links are interpreted as representing favorable, bilateral
relationships between the parties. In the model, two players are linked if and only if each of them
independently expresses a desire to cooperate with the other. It is likely that such players will
work together toward mutually beneficial outcomes in the political process. The absence of a
link between two players means that at least one player excludes the other in her proposal for the
coalition government. It is easy to imagine that there will be sore feelings between unlinked
players and that such players will draw less benefit from being in the government. Hence, it is
natural to assign less weight to players with missing links.
By using the proposed method of dividing the surplus, the government formation game is
a potential game, as defined by Monderer and Shapley (1996). A useful fact about potential
games is that only a subset of the Nash equilibria of potential games coincides with the set of
strategy profiles that maximize the potential. This renders potential maximization useful as a
refinement tool.3 In fact, the government formation game that we study here is a special case of
a cooperation formation game. Garratt and Qin (2001) show that for three-player cooperation
formation games only two possible cooperation structures survive potential maximization, and
3 See Monderer and Shapley 1996, Section 5.
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that generically the cooperation structure selected by potential maximization is unique. Here the
possibilities are either a two-party government or a three-party government with full cooperation
(i.e., every pair of players is linked in the cooperation structure). We provide conditions for
either type of government. These are expressed in terms of the size of the government surplus
and the distances between parties on the policy line.
To assess the effectiveness of potential maximization as a refinement of Nash equilibrium
we test it experimentally. As reported in Monderer and Shapley (1996), potential maximization
has proven useful as predictor of play in other contexts. In particular, the equilibrium selection
predicted by potential maximization was supported by the experimental results of Van Huyck et
al. (1990), who considered the Stag Hunt game. More recently, Goeree and Holt (1998) and
Anderson, Goeree, and Holt (2001) show that observed play in minimum effort coordination
games could be accounted for by maximization of a stochastic potential function.
Experimental studies of coalition formation have a long history in the social sciences.
Some of the early experiments were conducted by political scientists (Laing & Morrison, 1973;
sociologists (Gamson, 1961, 1964), economists (Murnighan & Roth, 1977, 1978), and game
theoreticians (Kalisch et al., 1954: Maschler, 1965). Many of the early studies focused on
psychological aspects of coalition formation behavior, whereas others (e.g., Kalisch et al., 1952;
Maschler, 1965) were designed to test theoretical predictions of coalition formation modeled as
games in characteristic function form (see Kahan & Rapoport, 1984, for a review of this early
experimental literature).
We follow the later studies by focusing on games that are both nonzero sum and non-
superadditive. The nonzero-sum assumption is in accordance with Riker (1967) who states “the
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greater portion of political activity in forming coalitions is nonzero sum in the sense that
different minimal winning coalitions win different amounts and the loss to the loser may not
equal the gain to the winner.” Our focus on non-superadditive games reflects the idea mentioned
above that adding new members to an already winning coalition might decrease the value of the
coalition. Unlike Riker and Maschler, we do not assume that adding a new member to a winning
coalition depreciates its value to zero. Instead, we assign the grand coalition (which is the only
non minimal winning coalition in our design) a positive value. This allows for the selection of
non-minimal winning coalitions in our model, a feature that is borne out in the experimental
results reported below.
2. Model
Consider a parliamentary system with three parties, where each party has an exogenously given
policy position represented by a point on the real line. The policy position of each party i∈ N
={A,B,C} is denoted pi∈ℜ, where pA ≤ pB ≤ pC. Let p=(pA, pB, pC)∈ℜN denote a vector of policy
positions. Each party controls a number of seats that it won in an election. Seat shares are
assumed to be such that any two parties (or all three parties) can combine to control a majority of
the seats, but no single party has a majority. There is a surplus of G that is received by the
members of whichever coalition government forms. However, this surplus is reduced by an
amount determined by the distances between the policy positions of the coalition government
members and the position, y, chosen by the government. Specifically, the value to any coalition
is given by the characteristic function v(p,G) where
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≥
−−
=∑∈ℜ∈
otherwise
SifpyGGpSv Si
iy N
0
2)(max),;(
2
(1)
The characteristic function specified in (1) is the same as that used in Austin-Smith and Banks
(1988). We use it to define a government formation game in which parties A, B, and C are the
players. We restrict attention to parameter values for which v(S;p,G) > 0 whenever 2≥S . This
ensures that there is always some positive value to being able to form the government.
The strategy set of player i is denoted by }.\{ iNSi ⊆=Π The empty set is included in
iΠ . A strategy ii Π∈π is a set of parties with whom player i wishes to form a government. For
instance, the strategy set of player A is {=Π A ∅, B, C, {B,C}}, signifying that player A can
choose to cooperate with nobody, party B, party C, or both parties B and C. Let .iNi Π×=Π ∈
Each strategy profile Π∈= ),,( CBA ππππ determines a cooperation (link) structure g(π) in the
following way. Given π ∈ Π, a link between parties i and j forms if and only if the desire to
cooperate is mutual, i.e., if and only if ji π∈ and ij π∈ .4 Denote an undirected bilateral link
between parties i and j by ji : . Then }.:{)( ji jandijig πππ ∈∈= All parties that are
linked (directly or indirectly) in the cooperation structure that results from some strategy profile
played in the government formation game are assumed to form the coalition government.
Payoffs to each player under a given strategy profile are determined by the Myerson
values. The Myerson values for members of a coalition depend upon the characteristic function
v(p,G) and the graph g(π) resulting from the selected strategy profile, π. Given a cooperation
4 Recall from the introduction that links are interpreted as representing favorable, bilateral relationships betweenthe parties. Hence, it is natural to form links only when the desire to cooperate is mutual.
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structure g(π) and a coalition NS ⊆ , let )(/ πgS denote the partition of S into subsets of
players who are connected (directly or indirectly) by g(π) in S. To specify the Myerson value, let
),()( Gpv g π denote the characteristic function determined according to
∑∈
⊆=)(/
)( .),,;(),;(π
π
gSR
g NSGpRvGpSv
The value ),;()( GpSv g π reflects the fact that coalition S may not be able to form due to a lack of
connectedness among all the members of S. We denote the Myerson values by the vector
is the Shapley value for player i in the game ).,()( Gpv g π
The characteristic function v(p,G) specified in (1) is invariant to changes in scale of the
party positions. In words, the relative policy positions of the parties matter, whereas the absolute
policy positions do not. For this reason, and because we assume pA ≤ pB ≤ pC, the three
parameters pA, pB, and pC can be replaced with two distance parameters AB ppd −=1 and
.2 BC ppd −= The Myerson values for each cooperation structure are given in Table 1 as a
function of the parameters d1, d2, and G. In summary, the government formation game consists
of players’ strategy sets ΠA, ΠB, and ΠC, and their payoff functions )),,(()( GpvgA
πψ
)),,(()( GpvgB
πψ and )),,(()( GpvgC
πψ .Π∈π
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Table 1. Myerson values for each of the eight possible cooperation structures.
3. Theoretical Predictions
A variety of solution concepts apply to the government formation game. We consider Nash
equilibrium and some of its refinements. Because of the way payoffs are defined the government
formation game is a potential game (see Garratt and Qin, 2001). Hence, one refinement that we
consider is potential maximization. This refinement involves finding the set of strategy profiles
for the game that maximizes the potential of the game. This set of strategy profiles is meaningful
because it always constitutes a subset of the Nash equilibria for the game. Moreover, the
potential has the property that its value increases with beneficial unilateral deviations by the
players. Hence, individual actions taken to improve own welfare lead to higher values of the
potential. Potential maximization is a particularly effective refinement in this game, because it
always yields a unique prediction of the cooperation structure (and payoffs) that results from
play of the game.5 We also examine strong Nash equilibria (SNE), Coalition proof Nash
equilibria (CPNE), and solutions resulting from iterated elimination of weakly dominated
5 Further justification of potential maximization as a refinement of Nash equilibrium is provided in Garratt and Qin(2001).
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strategies (IEWDS). SNE and CPNE are less desirable as refinements since they may not be
unique and may not exist. Nevertheless, the comparisons are instructive.
3.1 Nash equilibria in weakly dominant strategies
The first result identifies parameter values for which each party has a weakly dominant
strategy to cooperate with the other two parties.6 The result is followed by a corollary that
identifies the parameter set for which full cooperation emerges as a Nash equilibrium in (weakly)
dominant strategies.
Proposition 1. Party A has a (weakly) dominant strategy to cooperate with parties B and C if
and only if 0),,( 21}:,:{ ≥GddCBCA
Aψ and .0),,( 21}:,:{ ≥GddCABA
Cψ
Party B has a (weakly) dominant strategy to cooperate with parties A and C if and only if
0),,( 21}:,:{ ≥GddCBBA
Aψ and .0),,( 21}:,:{ ≥GddCBBA
Cψ
Party C has a (weakly) dominant strategy to cooperate with parties A and B if and only if
0),,( 21}:,:{ ≥GddCBCA
Aψ and .0),,( 21}:,:{ ≥GddCABA
Cψ
See Appendix A for proof. Checking whether an individual has a dominant strategy to cooperate
fully involves checking the appropriate two conditions outlined in Proposition 1. However,
checking whether all three players have a dominant strategy to cooperate fully does not involve
checking six conditions. This is because two-thirds of the six individual conditions obtained from
applying Proposition 1 to each of the three parties are redundant. To check that all three parties
have a dominant strategy to cooperate fully it is sufficient to check only two conditions.
6 A weakly dominant strategy for a player is one that is at least as good as any other strategy she can choose,regardless of the strategy choices of the other players.
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Corollary 1. All three parties have a (weakly) dominant strategy to cooperate with both of the
other parties if 0),,( 21}:,:{ ≥GddCBCA
Aψ and .0),,( 21}:,:{ ≥GddCABA
Cψ
The corollary is true because the conditions for party B to have a (weakly) dominant
strategy to cooperate with the other two parties are implied by the conditions provided for either
party A or B.
Graphical representation: Fix G. The conditions 0),,( 21}:,:{ ≥GddCBCA
Aψ and
0),,( 21}:,:{ ≥GddCABA
Cψ describe a rotated hyperbola in (d1, d2) space. The intersection of the two
graphs in 2+ℜ defines the set of parameter values for which every party has a dominant strategy
to cooperate with both of the other parties (See Fig. 1).
3.2 Potential maximization
We adapt the definition of potential from Monderer and Shapley (1996). A potential for a
government formation game (with parameter values ),, 21 Gdd is a function ℜ→Π:P such that
for any i ∈ N, π ∈ Π, and ,’ii Π∈π ).(),(),,(),,( ’
21)(
21),( ’
πππψψ πππ PPGddGdd iigi
gi
ii −=− −−
The predictions of potential maximization are described below.
Proposition 2. A two-party government will maximize the potential if and only if either
0)),(( ≤GpvAφ or .0)),(( ≤GpvCφ A three-party government with complete cooperation will
maximize the potential if and only if 0)),(( ≥GpvAφ and .0)),(( ≥GpvCφ
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Proof: Given the specification of the characteristic function in (1) and since pA ≤ pB ≤ pC, the
inequalities )),(()),(( GpvGpv BA φφ ≤ and )),(()),(( GpvGpv BC φφ ≤ hold. The result then
follows from Remark 3(ii) of Garratt and Qin (2001). ó
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Figure 1. Dominant strategies, potential maximization, and the five games.
Proposition 3. A two-party government resulting from potential maximization will include the
two parties that are closest together on the policy line.
Proof: Suppose 0)),(( <Gpviφ for some i ∈ {A,C} so that by Proposition 2 some two-party
government forms. Since pA ≤ pB ≤ pC, it must be the case that either parties A and B are closest
Potential maxis completecooperation
Each player hasa dominantstrategycooperate withthe other twoplayers.
Potential maxis parties Aand B
Potential maxis parties Band C
d2 =pC – pB
d1 = pB – pA
1
3
2
5
4
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together on the policy line or parties B and C are closest together. Due to the obvious similarities
we only prove the case where ).()( BCAB pppp −<− For any two-party coalition {i,j}, i,j ∈ N,
the solution to the maximization problem in (1) is .2/)(* ji ppy += Consequently,
2)(2
1}),({ ji ppGjiv −−= for any i,j ∈ N. It is immediate that },2:)(max{}),({ == SSvBAv
and hence by Remark 3(i) in Garratt and Qin (2001) a two-party government including parties A
and B forms. ó
Graphical representation: Fix G. Then each of the conditions in Proposition 2, 0)),(( =GpvAφ
and 0)),(( =GpvCφ describe a rotated ellipse in (d1,d2) space. The intersection of the two
ellipses in 2+ℜ defines the set of parameter values for which the potential maximizing outcome is
full cooperation (See Fig. 1).
Although the result in proposition 3 is not surprising, it may be taken as an indication of
the reasonableness of the solution concept. Some well-known, sequential processes of coalition
formation, such as the method described in Austin-Smith and Banks (1988), do not have this
desirable property.
It is perhaps surprising that the magnitude of G matters in determining whether the
government will be two-party (minimal winning) or three-party (non minimal winning). This
occurs because the Myerson value averages marginal contributions that occur when a party
enters a coalition that is already winning with cases where the coalition is not already winning.
Whenever a party joins a coalition that is already winning, it reduces the surplus available to the
coalition members. Whether or not such instances are overridden by cases where the party is
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instrumental in causing a coalition to be winning depends on the magnitude of G. The practical
implication is that we expect larger governments to be more prevalent in instances where the
gains to being in power are large.
4. Treatment Parameters
We constructed five games to be studied experimentally. The parameters selected for the five
games are shown graphically in Fig. 1. The games were selected to test the predictions of
Propositions 2 and 3. They vary from one another in their parameter values. Games 1, 4, and 5
lie in the intersection of the set of parameter values for which each party has a weakly dominant
strategy to cooperate with the other two parties and the set of parameter values for which the
unique potential maximizing strategy profile is complete cooperation. These games differ in
terms of the degree of symmetry: all three parties are seen to have the same party position in
Game 1, only parties A and B are the same in Game 4, and all three parties have different policy
positions in Game 5. Game 2 lies in the set of parameters for which potential maximization
predicts a two-party coalition government including parties A and B. Game 3 lies in the set of
parameter values for which the unique potential maximizing strategy profile is complete
cooperation, but unlike games 1, 4, and 5, it lies outside the set of parameter values for which
each party has a dominant strategy to cooperate with the other two parties.
With three parties there are eight possible cooperation structures. Appendix B illustrates
the eight possible cooperation structures for each of these five games and the associated payoffs
for each of the three players. Numbers in parentheses in Appendix B indicate negative payoffs
(losses). Observe that in Games 1-4 the efficient, minimal winning government is the coalition
{A,B}, whereas in Game 5 it is the coalition {B,C}. Once again, potential maximization predicts
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complete cooperation (the non-minimal winning government with all parties directly linked) in
all but Game 2.
4.1 Other solution concepts
In section 3 we produced a general theory for checking (weakly) dominant strategies and
potential maximization. These notions select a subset of the Nash equilibria for each game. In
this section we also evaluate other solution concepts that apply to the games in Fig. 1 on a case-
by-case basis. These include SNE and CPNE, two well-known refinements of Nash equilibrium.
CPNE is defined recursively. As compared to SNE, there is an additional requirement that a
deviation by a coalition must be valid in the sense that no proper subcoalition would want to
deviate from the deviation. In this case, any proper subcoalition of a two-player coalition is a
singleton. Thus, any deviation by a two-player coalition is valid provided that deviation
guarantees individual rationality. In our games, single-player coalitions receive a payoff of zero
and all two-player coalitions yield positive payoffs for both players in the coalition (see
Appendix B). Thus, SNE and CPNE coincide for our games. In addition, we consider solutions
obtained by IEWDS. This solution concept is not a refinement of Nash equilibrium. Moreover,
its predictions depend on the order in which strategies are eliminated.
4.2 Predictions
Tables 2a-2c present the predictions of the solution concepts discussed above for each of the five
games. With four strategies for each of the three players, there are 64 strategy profiles for each
game. Only Nash equilibrium strategy profiles are listed for each game. In the table, “Weak
D.S.” means that each of the strategies in the strategy profile is a weakly dominant strategy;
“Pot. Max.” means potential maximizing strategy profile; “IEWDS” means the strategy profile
survives iterated elimination of weakly dominated strategies (for any iteration); and
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“SNE/CPNE” means the strategy profile is both a strong Nash and a coalition proof Nash
equilibrium.
Potential Maximization. Tables 2a-2c show that the strategy profile ({B,C},{ A,C},{ A,B})
is the unique potential maximizer in Games 1, 3, 4, and 5. In Game 2, every strategy profile that
gives rise to the cooperation structure A:B (there are nine) is potential maximizing.
IEWDS. The profile ({B,C},{ A,C},{ A,B}) is the sole survivor of IEWDS in Games 1, 4,
and 5. It is one of the eight Nash equilibria of these games. There are four strategy profiles in
Game 2 that survive IEWDS, namely, (B,A,∅), (B,{A,C},∅), ({B,C},A,∅), and
({ B,C},{ A,C},∅). Each of these four strategy profiles is a Nash equilibrium. Moreover, each
gives rise to the cooperation structure A:B, which yields the same payoff of 270 to parties A and
B (Fig. 2b). Party C cooperates with nobody in each of the strategy profiles that survive IEWDS.
There are eight strategy profiles in Game 3 that survive IEWDS. Of these eight profiles, five are
Nash equilibria, namely, (B,A,∅), (B,{A,C},∅), ({B,C},A,∅), (B,A,{A,B}), and
({ B,C},{ A,C},{ A,B}), and three are not. Four of the Nash equilibrium strategy profiles that
survive IEWDS in Game 3 give rise to the cooperation structure A:B (with symmetric payoffs to
players A and B, see Fig. 2c), and the fifth gives rise to the cooperation structure A:B,B:C,A:C.