Formulation and Applications of Hierarchy Games 1 ...

Formulation and Applications of Hierarchy Games Mitsutaka Matsumoto Department of Value and Decision Science, Tokyo Institute of Technology, Japan. (E-mail: [email protected]) Abstract In this paper, I discuss the formulation and some applications of a framework called \hierarchy games". Hierarchy games are a game theoretical framework in which two or more games interact with one another.

The interaction among games can be best described by the following two

characteristics: 1)the outcome of a lower-level game could restrict the available actions of a player in higher-level. 2)the payo of a higher-level player is shared by lower-level players. The above ideas were rst proposed by Hausken in 1995.

The paper attempts to develop further those

ideas. First, some parts of the framework are de ned more rigorously than Hausken's formulation. Secondly, some applications of the framework to the eld of International Relations are discussed.

1 Introduction 1.1 Individual rationality and group rationality In a \social situation" where two or more actors seek their own goals, possibilities of con icts, competition or cooperation among actors are inherent. To reveal the various logics underlying con icts and cooperation among actors in a social situation is an important task of social sciences such as politics, economics, sociology, ethology and so on. Considering such problems, it is important to recognize the inconsistencies between individual rationality and group rationality.

Game theory explains such inconsistencies clearly and simply,

which is why it is widely used in various elds of social sciences. The \prisoner's dilemma" model represents the above inconsistencies most simply. various frameworks to analyze the inconsistencies more elaborately have been proposed.

By now, Among

them, the \theory of repeated games" and the \evolutionary game theory" would be the most typical ones.

We can add to these the \hypergames" which assume situations with incomplete

information([Bennett 80]), or \the institutional approach" which focuses on the formation of institutions as a means to seeking actors' group rationalities as well as individual rationalities([Okada 93]). This paper looks into the interactions among social situations, and aims to develop a framework which is called \hierarchy games" or \multi-level games". The hierarchy games framework considers a situation in which a set of games form a hierarchy. The games interact with one another so that each game aects a player of an one-level higher or lower game. International politics is an example of how the outcomes of a higher-level game played between nations can in uence the lower-level game between actors within each nation. I show that this framework oers another approach to solving the prisoner's dilemma. A simple example which I named \two-level prisoner's dilemma" is presented in Section 3 and analyzed in Section 5. This model illustrates the logic.

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1.2 Application to international politics This paper attempts to apply the hierarchy games framework to international politics as targets of applying the framework.

The reason why I chose international politics is that this eld has

attached great importance to the viewpoints of \interactions among social situations". The traditional methodology of international politics considers nations as actors of international societies (\the third image" in [Waltz 54]) and analyzes individual and group rationalities. However, in the real world, individually rational actions are not always taken by nations. In order to explain such phenomena, interactions among games have been discussed as are shown in Section 2.

A second example is

presented in Section 5. The model which I named \the dilemma of free trade" illustrates one of the logics which yields such phenomena.

2 Related work In international politics, the topic \interactions among social situations" has been discussed in two contexts. One is \issue-linkages", the other is \two-level games". An issue linkage indicates a situation where the same actors are engaged in two or more social situations. An example is the United States and the Soviet Union being engaged in one game to secure a strategic advantage in the Middle East and another game to secure a strategic advantage in Central America. Alt and Eichengreen formulated such a situation game theoretically as \parallel games" ([Alt and Eichengreen 89]). They also de ned \overlapping games" as those where one player is con ned to adopting one strategy but is engaged in two games, each of which involves a separate rival. Inohara's \integration of games" is similar to this concept ([Inohara et al 97]). The hierarchy games framework is in the stream of \two-level games". proposed by Putnam ([Putnam 88]).

The concept was rst

In the analysis of international negotiations, he considered

not only the international level but also the national level where domestic groups pressure their governments. However, his formulation was not so mathematical. \Multi-level games" proposed by Axelrod and Keohane ([Axelrod and Keohane 93]) and \nested games" by Tsebelis ([Tsebelis 90]) de ne games with variable payos where the payos are in uenced by other games.

So these approaches cover both issue-linkages and two-level games. However, in

their formulations it is not clear what factors actually in uence the payos. Hausken argued in his \hierarchy games" that games interact through \action restrictions" and \payo distributions". In these points, my formulation follows his.

But he did not de ne explicitly what factors determine

\action restrictions" and \payo distributions", and this is where my formulation goes a step further than his.

3 The two-level prisoner's dilemma In this section, I show a simple example which I named the two-level prisoner's dilemma game to illustrate the concepts of hierarchy games prior to describing the formulation of the framework in the next section.

Structure of model

Let us consider a simple society model where there exist two nations, P5 and

P6, each of which consists of two domestic groups (Figure 1). Nation P5 consists of P1 and P2. P6 consists of P3 and P4. Now we have three societies | two domestic societies, Game1, Game2, and an international society, Game3. Let us assume that all of the three games are prisoner's dilemmas.

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Game3 P6

P5

Game1

Game2 P1

P2

P3

P4

Figure 1: The two-level prisoner's dilemma game

C

C D

D

R1 ; R1 S1 ; T1 T1; S1 P1 ; P1

T > R1 > P1 > S1 )

( 1

Table 1: The payo matrix of Game1 and Game2

Payo matrices

Let's assume Game1 and Game2 are identical prisoner's dilemma games whose

payo matrices are as Table 1. But for Game3, it is slightly dierent from a usual prisoner's dilemma in that the players have a medium action M, besides usual C action and D action. The payos of M actions are an average of those of C and D (Table 2).

Interactions among games

So far, three independent games were de ned. Now I describe the

interaction among the games, which is the main point of the framework. There are two elements in the interaction. Firstly, in this context, the domestic outcomes restrict the nation's available actions, which means that the nations do not always have all the three actions | C, M and D. The scope of available actions depends on the outcome of its domestic game. Let us assume like this: The scope of a nation's available actions depends on the aggregate payo of its domestic players or GDP of the nation. This setup is

S6 have the same form of relationship. The second element of interaction is the payo distribution. Suppose x is the payo of P5 in Game3. As x is payo of P5 itself, P5 tries to maximize x. But, at the same time, x is shared by P1

as Table 3

1 . We assume that Game2 and

and P2. This is the same for P6. P6's payo is shared by P3 and P4. So the payo of a domestic player is the sum of its payo in the domestic game and the payo distribution from its nation. The ratio of the distribution depends on the outcome of Game3. Here, for simplicity, I assume that it is distributed equally to domestic players whatever the outcome is.

Summary

In this model, due to the interaction with the higher-level game, the domestic game play-

ers are given incentives to cooperate so that their nation can take a better action in the international game. The Nash equilibria are analyzed in Section 5. 1 One question might occur. Why is it restricted to C when GDP is minimum? Not to D? This setup is naturally understandable if we think of the tragedy of a common land model instead of the prisoner's dilemma. In the common land tragedy model, the cooperation corresponds to a small number of sheep pastured. So even if a nation wants to pasture many sheep, if its GDP is small, it can not, which means that when GDP is small, the nation can pasture only small number of sheep. It corresponds to the situation that the nation can only take C action.

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C

C M D

M

R3 ; R3

R S ; 12 (R3 +T3 ) T P3 +R3 +S3 ) T P R3 +S3 ), 1 1 2 (T3 +P3 ); 2 (S3 +P3 )

1 2 ( 3 + 3) 1( + 4 3 1( + + 3 3 4

R S R T

1( + ) 3 2 3 1 ( + ), 3 3 2

T3 ; S3

D

S3 ; T3

T P S P

1 2 ( 3+ 3) 1 ( + ), 3 2 3

P3 ; P 3

T > R3 > P3 > S3 )

( 3

Table 2: The payo matrix of Game3

P1

P2

C

C

C

D

D

C

D

D

S5

fC,M,Dg fC,Mg fC,Mg fCg

Table 3: The interaction between Game1 and P5 (action restriction)

Some applications We are able to nd social situations which have the same logic as the two-level prisoner's dilemma.

An example is from social thoughts of Rousseau. Rousseau described in \A Discourse on the Origin of Inequality" that once a social contract were established by some group, then other people had to form their own social contracts in order to oppose the eciency of using resources and labors by people under the rst social contract. This indicates that a higher-level game (which are played among social contracts) gives incentives of cooperation to people under each contract. Another example is from international politics of these decades. The process and the impacts of emergence of regional regimes such as EU, NAFTA and APEC are one of the hot topics in the eld of international relations. The model above corresponds to a phenomenon, for example, the foundation of NAFTA was accelerated by EU's integration. The framework provides a mathematical tool for such analysis.

4 Formulation of hierarchy games The framework of hierarchy games is formalized in this section. Prior to formalization, some premises are made.

4.1 Premises

Premise 1

A hierarchy of games is a tree structure.

Premise 2

Each game (each node of the tree) is a strategic game.

Premise 3

Each edge of the tree represents an interaction. A game (a node) interrelates with the

parent game (parent node) by correlating with a player in the parent game. correlate with only one child game.

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A player can

P5

P6

P1 : P2

C

C

05:05

C

D

D

C

D

D

: : 0:5 : 0:5 0:5 : 0:5 0:5 : 0:5

Table 4: The distribution of P5's payo to P1 and P2

Premise 4

An outcome of a lower level game determines the range of actions available to its parent

player.

Premise 5

If a player has its child game, the payo of the player is distributed to the players of the

child game.

Premise 6

The ratio of payo distribution from a parent player is determined by the outcome of

the parent game.

4.2 Formulation of hierarchy games (augmented strategic game form)

J; fGg)

3=(

(1)

J is a tree which represents the hierarchy. 2. fGg is a set of strategic games. Each corresponds to a node in the tree. Each element Gx 2 fGg is a strategic game. In order to represent the interrelation with a higher 1.

level game, ve components are added to the usual three components.

Gx = (Nx; fSigi2Nx ; ffigi2Nx ; Gs ; psj ; gx ; Yx ; Zx )

1. 2. 3.

4.

5. 6.

(2)

Nx is the set of players. Si is the set of actions available to player i. fi : S ! R is the payo function of player i, where S = S1 2 1 1 1 2 Sn. (Besides the payo in Gx , player i gets a distribution from the higher-level game player, which is mentioned below.) Gs: Gx 's parent game (parent node). If Gx is the highest level game, Gs is empty. psj is a player in Gs . psj interrelates with Gx . gx is the payo distribution function. A function from Sps1 2 1 1 1 2 Spsn (the outcomes of Gs ) to jNx j-dimension vector. The sum of elements is 1, as the vector represents a ratio. Table 4 is an example.

7.

Yx represents Gx's action restriction to psj . A function from S1 2 1 1 1 2 SjNxj to 2Ssj (all subsets of Ssj ). Table 3 is an example.

8.

Zx is the set of players in higher levels, who know the outcome of Gx .

When a hierarchy game is represented in extensive form (following subsection), this determines its information partition.

5

C MD

CM

CM

C

P6 P5

C MD

P4

C

CM D

C

C

D

D

CM

C D

C D

C

C

C D

D

C

C D

C D

D

C

C

D

C D D

P3 C

D

P2 C

D

P1

Figure 2: The two-level prisoner's dilemma in extensive form

Game3

Game1

Game2

Figure 3: The tree of the two-level prisoner's dilemma game

4.3 Extensive game representation An extensive game represents a hierarchy game, too. The ve components of the extensive games

K:

(

P : player partition, p: probability distribution of chance moves, U : information h: payo) can be determined using the information represented by the augmented strategic

the game tree,

partition,

form. The two-level prisoner's dilemma game is represented as Figure 2 where the payos are omitted.

5 Analysis of the two-level prisoner's dilemma game In this section, I describe the two-level prisoner's dilemma game formally and analyze its Nash equilibria.

5.1 Representation 

J; fG1 ; G2 ; G3 g)

3=(

J is Figure 3. G1 = (fP 1; P 2g; f(C; D); (C; D)g; ff1 ; f2 g; G3 ; P 5; g1 ; Y1 ; fP 5g) ff1 ; f2 g is Table 1. g1 is Table 4. Y1 is Table 3. G2 = (fP 3; P 4g; f(C; D); (C; D)g; ff3 ; f4 g; G3 ; P 6; g2 ; Y2 ; fP 6g) f ,g,Y are identical forms with those of G1. G3 = (fP 5; P 6g; f(C; M; D); (C; M; D)g; ff5; f6 g; ) The hierarchy game tree

  

5.2 Nash equilibria of the game

Dominating actions of P5 and P6 1. If the actions available are

fD,M,Cg， D

2. If the actions available are

fM,Cg， M

3. If the actions available are

fCg， C

is the dominating action.

is the dominating action.

must be taken.

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! # !

P3,P4 P1

P2

C

D

(C, C) C (1)

(C,D) or (D,C) D (2)

R1 + P23

S1 + P3 +4 S3 P1 + S23

T1 + P3 +4 S3

C (3)

(D, D)

D (4)

R1 + T3 +4 P3

T1 + T3 +P3 +8R3 +S3

S1 + T3 +P3 +8 R3 +S3 P1 + R3 +4 S3

C (5)

R1 + T23

T1 + T3 +4R3

D (6)

S1 + T3 +4 R3 P1 + R23

Table 5: Payo matrix of lower game players

(1) CCCC DDDD CCCD CDDD (CC)(DD) (CD)(CD)

(2)

(3)

(4)

(5)

(6)

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

_1 ^1 ^1 _1 _1 ^1 ^1 _1 ^1 _1 ^1 _1 * *

*

*

*

*

*

*

*

Table 6: Necessary and sucient condition for each outcome to be a Nash equilibrium

Actions of P1,P2,P3 and P4 I assume that P5 and P6 take the dominating actions. Table 5 is the payo matrix whose elements are P1's payos. As the payo structure is symmetric, other players' payos are omitted. The numbers (1),..,(6) in the table indicate the column numbers. There are six types of outcomes for the four players:

1)

`CCCC': all four players cooperate;

2) `DDDD': all defect; 3) `CCCD': three of the four cooperate; 4) `CDDD': only one of the four cooperates; 5) `(CC)(DD)': two cooperate. The two are from the same nation; 6) `(CD)(CD)': two cooperate. The two are from dierent nations.

Proposition 1 Each of the six outcomes can be a Nash equilibrium. Table 6 shows the necessary and sucient condition for each of the six outcomes to be a Nash equilibrium.

The six columns correspond to the six columns of Table 5.

The numbers (1),..,(6)

identify the correspondences. In Table 6, the `_1' indicates that the C action payo of the column in Table 5 is greater than or

equal to the D action payo. The `^1' indicates that the C action payo is less than or equal to the D action payo. The `*' indicates \don't care". For example, the fourth row of Table 6 means that `CDDD' is a Nash equilibrium if and only if:

S1 + T3+P3+8R3 +S3  P1 + R3 +4 S3 (fourth column) T T +R and R1 + 23  T1 + 3 4 3 ( fth column) T + R R and S1 + 3 4 3  P1 + 23 (sixth column). Proposition 2 Each of the six outcomes can be a u_ n_ i_q_ u_ e_ Nash equilibrium. Table 7 shows the necessary and sucient condition for each combination of actions to be unique Nash equilibrium. The `_' indicates the C action payo is greater than the D action payo. The `^' indicates the C action payo is less than the D action payo. The outcomes `(CC)(DD)' and `(CD)(CD)' have three cases, respectively, which make the outcomes become unique Nash equilibrium.

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(1)

_ ^ ^ ^ ^ ^ ^ ^ ^ ^

CCCC DDDD CCCD CDDD (CC)(DD)

(CD)(CD)

(2)

_ _ ^ ^ ^1 ^ _ ^ _ *

(3)

_ ^ _ ^ _ ^ _ ^ ^1 ^

(4)

_ ^ _ ^ _ ^ ^ _1 _ _

(5) *

^ _ ^ _1 _ _ _ ^ ^

(6)

_ ^ _ _ _ _ _ _ _ _

Table 7: Necessary and sucient condition for each outcome to be unique Nash equilibrium

6 The dilemma of free trade model In this section, I show another example of the hierarchy games.

6.1 The dilemma of free trade According to the theory of comparative advantages in international economics, international division of labor increases welfare of all the nations. While the statement is true theoretically, in the real world, trade liberalization has not proceeded smoothly, or sometimes it declines. Some researchers argue that the most important cause is that the societies are divided between those who bene t and those who lose by greater specialization, while it bene ts a nation as a whole (e.g. [Rogowski 89]). This is a typical example of phenomena where nations do not choose individually rational actions in the international game and thus we need to consider domestic politics. The simple model presented in this section represents the logic above.

6.2 The hierarchy game model Let us consider, again, a situation where two nations exist and each nation consists of two groups or individuals.

The hierarchy game

J , which is identical to Figure 3. 3 = (J; fG1 ; G2 ; G3 g)

We use a hierarchy game tree,

The international game

G3 = (fP 5; P 6g; f(F; P ); (F; P )g; ff5 ; f6 g; ) The F corresponds to a free trade policy, while P represents a protectionist trade policy. Let the payo matrix ff5 ; f6 g be as follows. F P

F

P

5,5

3,2

2,3

2,2

8

In some models, a game is set up such that the payo is maximum when a nation takes a protectionist trade policy and the other nation takes a free trade policy. But, here, we assume that free trade is achieved only when both countries take free trade policy, which means that a unilateral protectionist policy does not bene t a nation, or it might require a cost. So we assume that the free

F; F ) is the Nash

trade policy is the dominating action for each nation. In the payo matrix above, ( equilibrium and it is Pareto ecient.

Domestic games

f , g, Y

Both nations have domestic games of the same form.

G1 = (fP 1; P 2g; f(L; V ); (L; V )g; ff1 ; f2 g; G3; P 5; g1; Y1 ; fP 5g) G2 = (fP 3; P 4g; f(L; V ); (L; V )g; ff3 ; f4 g; G3; P 6; g2; Y2 ; fP 6g) are the same forms.

Let the payo matrix

ff1 ; f2g

be as follows.

L V

L

V

1,1

1,0

0,1

0,0

L is the action to leave the policy decision to the government (P5), while V

is the action to veto

the free trade policy. The table re ects a setup that the veto action itself has a bad reputation and does not result in a good payo.

L is the dominating action for both players, and (L; L) is the Nash

equilibrium. The following table represents g1 . Only when both P1 and P2 choose L, P5 F and P as its available actions. Otherwise, P5 has only P . The g2 is the same form as g1. S5 P1 P2 L L fF; P g L V fP g V L fP g V V fP g

Action restrictions has both

Payo distributions

Based on the logic of the dilemma of free trade, we use the table below as

the payo distribution function

Y1 . Y2 is the same form. P5

F F P P

P6

P1 : P2

F P F P

2:

01

: : : : 0:5 : 0:5 05:05 05:05

When the free trade is achieved and the nation's payo is 5, the payo is distributed such that one person gets 10 and the other gets -5. When free trade is not achieved and the payo of the nation is 2 or 3, then the payo is distributed equally to the two domestic groups.

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6.3 Analysis

Dominating actions of P5 and P6 Following is the case of P5. P6 is the same form. 1. When both P1 and P2 choose

L and

the actions available to P5 are

fF; P g,

and the actions available to P5 is

fP g,

then

F

is the

dominating action of P5. 2. When at least one of P1, P2 chooses

P.

V

P5 has to choose

Actions of P1,P2,P3 and P4 We assume P5 and P6 take the dominating actions. The total payos of P1 and P2 are as follows. Payos of P3 and P4 are the same form. 1. When P6 chooses

F: P1

n

L V

L

P2

V

11,-4

2,1

1,2

1,1

L; V ) is the Nash equilibrium. When P6 chooses P :

The ( 2.

P1

n

L V

P2

L

V

2.5,2.5

2,1

1,2

1,1

L; L) is the Nash equilibrium.

The (

Conclusion

L; V; L; L) and (L; L; L; V ).

So the Nash Equilibria for P1, P2, P3 and P4 are, (

In the former equilibrium, P2 vetoes, while in the latter equilibrium, P4 vetoes. So in both cases, one of the nations takes the protectionist trade policy, and therefore, the free trade is not achieved in either case. So, even though the free trade policy is the dominating action of each nation in the international game and the veto to the free trade policy is the dominated action of each individual, the trade liberalization is impeded through the interaction between the games.

7 Conclusion and future research agenda In this paper, I have proposed a hierarchy games framework as a means to analyze the inconsistencies between individual rationality and group rationality. I have also improved the formulation of hierarchy games. On the \two-level prisoner's dilemma" model, I have shown that the interaction among games could give incentives to cooperate and make players overcome the prisoner's dilemma. This has been con rmed by the fact that `CCCC' can be an equilibrium under certain conditions. In the two-level prisoner's dilemma model, other equilibria besides `CCCC' may also draw the attention of researchers. For example, `CDDD' in which only one player cooperates and the other 10

three defect, also can be an equilibrium. It is up to future work to apply the equilibria to real life social phenomena. The \dilemma of free trade" model is one of the essential methods of research in the eld of international politics. In the future, I plan to apply this framework to concrete social phenomena. Game theoretically, my future work will be to investigate the patterns that emerge when equilibrium points of games change as a result of interactions among games.

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Economics and Politics 1, pp119-144, 1989.

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[Bennett 80] P.G.Bennet, \Hypergames:

developing a model of con ict,"

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pp489-507,

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Rationality and Society, Vol.7,

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[Putnam 88] Robert D. Putnam, \Diplomacy and Domestic Politics { The Logic of Two-Level Games,"

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[Rogowski 89] Ronald Rogowski, \Commerce and Coalition: How Trade Aects Domestic Political Alignments," Princeton University Press (1989). [Tsebelis 90] George Tsebelis, \Nested Games: Rational Choice in Comparative Politics," University of California Press (1990). [Waltz 54] Kenneth N. Waltz, \Man, the State and War | a theoretical analysis," Columbia University Press (1954).

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