Hedonic Diversity Games

03/01/2019 ∙ by Robert Bredereck, et al. ∙ 0

We consider a coalition formation setting where each agent belongs to one of the two types, and agents' preferences over coalitions are determined by the fraction of the agents of their own type in each coalition. This setting differs from the well-studied Schelling's model in that some agents may prefer homogeneous coalitions, while others may prefer to be members of a diverse group, or a group that mostly consists of agents of the other type. We model this setting as a hedonic game and investigate the existence of stable outcomes using hedonic games solution concepts. We show that a core stable outcome may fail to exist and checking the existence of core stable outcomes is computationally hard. On the other hand, we propose an efficient algorithm to find an individually stable outcome under the natural assumption that agents' preferences over fractions of the agents of their own type are single-peaked.

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1. Introduction

At a conference dinner, researchers split into groups to chat over food. Some junior researchers prefer to stay in the company of other junior researchers, as they want to relax after a long day of talks. Some senior researchers prefer to chat with their friends, who also happen to be senior researchers. But there are also junior researchers who want to use the dinner as an opportunity to network with senior researchers, as well as senior researchers who are eager to make the newcomers feel welcome in the community, and therefore want to talk to as many junior people as possible.

This example can be viewed as an instance of a coalition formation problem. The agents belong to two types (senior and junior), and their preferences over coalitions are determined by the fraction of agents of each type in the coalition. This setting is reminiscent of the classic Schelling model (Schelling, 1969), but there is an important difference: a standard assumption in the Schelling model is homophily, i.e., the agents are assumed to prefer to be surrounded by agents of their own type, though they can tolerate the presence of agents of the other type, as long as their fraction does not exceed a pre-specified threshold. In contrast, in our example some agents have homophilic preferences, while others have heterophilic preferences, i.e., they seek out coalitions with agents who are not like them.

There is a very substantial body of research on homophily and heterophily in group formation. It is well-known that in a variety of contexts, ranging from residential location (Schelling, 1969; Zhang, 2004a, b) to classroom activities and friendship relations (Moody, 2001; Newman, 2010), individuals prefer to be together with those who are similar to them. There are also settings where agents prefer to be in a group with agents of the other type(s): for instance, in a coalition of buyers and sellers, a buyer prefers to be in a group with many other sellers and no other buyers, so as to maximize their negotiating power. Aziz et al. (2014); Aziz et al. (2017) model this scenario as a Bakers and Millers game, where a baker wants to be in a coalition with many millers, whereas a miller wants to be in a coalition with many bakers. However, there are also real-life scenarios where agents can have different attitudes towards diversity: this includes, for instance, language learning by immersion (with types being learners’ native languages), shared accommodation (with types being genders), primary and secondary education (with types being races and income groups), etc. In all these settings we expect the agents to display a broad range of preferences over ratios of different types in their group.

Our contribution  The goal of our paper is to provide strategic foundations for the study of coalition formation scenarios where each agent may have a different degree of homophily. More concretely, we consider settings where agents are divided into two types (blue and red), and each agent has preferences regarding the fraction of the agents of her own type, which determine her preferences over coalitions. For most of our results, we assume that agents’ preferences are single-peaked, i.e., each agent has a preferred ratio of agents of her own type, and prefers one fraction to another fraction if is closer to than is. Our model allows agents to express a variety of preferences including both complete homophily and complete heterophily.

We model this setting as a hedonic game, and investigate the existence of stable outcomes according to several established notions of stability, such as core stability, Nash stability and individual stability (Bogomolnaia and Jackson, 2002; Banerjee et al., 2001).

We demonstrate that a core stable outcome may fail to exist, even when all agents have single-peaked preferences. Moreover, we show that deciding whether a core stable outcome exists is NP-complete. However, we identify several interesting special cases where the core is guaranteed to be non-empty.

We then consider stability notions that are defined in terms of individual deviations. There is a simple reason why a Nash stable outcome may fail to exist. However, we propose an efficient algorithm to reach an individually stable outcome, i.e., an outcome where if some agent would like to deviate from her current coalition to another coalition, at least one agent in the target coalition would object to the move. Our proof employs a careful and non-trivial adaptation of the algorithm of Bogomolnaia and Jackson (2002) for single-peaked anonymous games. Our algorithm is decentralized in the sense that, by following a certain set of rules, the agents can form a stable partition by themselves.

Related work  Our work is related to the established line of research that studies the impact of homophily on residential segregation. The seminal paper of Thomas Schelling (Schelling, 1969) introduced a model of residential segregation in which two types of individuals are located on a line, and at each step a randomly chosen individual moves to a different location if the fraction of the like-minded agents in her neighborhood is below her tolerance ratio. With simple experiments using dimes and pennies, Schelling (1969) found that any such dynamics almost always reaches a total segregation even if each agent only has a mild preference for her own type.

Following numerous papers empirically confirming Shelling’s result (Farley et al., 1993; Massey et al., 1994; Emerson et al., 2001; v. d. Laan Bouma-Doff, 2007; Clark and Fossett, 2008; Clark, 2008; Alba and Logan, 1993), Young (1998) was the first to provide a rigorous theoretical argument, showing that stability can only be achieved if agents are divided into homogeneous groups. In contrast, Brandt et al. (2012) showed that with tolerance parameter being exactly , the average size of the monochromatic community is independent of the size of the whole system; subsequently, Immorlica et al. (2017) extended this analysis to the two-dimensional grid. The recent work of Chauhan et al. (2018) considers a more general utility function, taking into account both the fraction of the like-minded agents in the neighborhood and the location in which each agent is situated; they investigate the existence of stable outcomes for several topologies, such as a grid and a ring.

We note, however, that our model is fundamentally different from Shelling’s model, for at least three reasons. First, we do not assume any underlying topology that restricts coalition formation. Second, the coalitions in an outcome of a hedonic game are pairwise disjoint, while the neighborhoods in the Schelling model may overlap. Finally, as argued above, our model does not assume homophilic preferences.

There is also a substantial literature on stability in hedonic games, starting with the early work of Bogomolnaia and Jackson (2002). Among the various classes of hedonic games, two classes are particularly relevant for our analysis: fractional hedonic games (Aziz et al., 2014; Aziz et al., 2017) and anonymous hedonic games (Bogomolnaia and Jackson, 2002).

In fractional hedonic games the agents are located on a social network, and they prefer a coalition to a coalition if the fraction of their friends in is higher than in . The Bakers and Millers game is an example of a fractional hedonic game, where it is assumed that each baker is a friend of each miller, but no two agents of the same type are friends. Aziz et al. (2017); Aziz et al. (2014), and, subsequently, Bilò et al. (2014, 2015) identify several special cases of fractional hedonic games where the set of core stable outcomes is non-empty. In particular, Aziz et al. (2014); Aziz et al. (2017) give a characterization of the set of strictly core stable outcomes in the Bakers and Millers game.

In anonymous hedonic games agents’ preferences over coalitions depend on the size of these coalitions only. Similarly to our setting, it is known that with single-peaked anonymous preferences, there is a natural decentralized process to reach individual stability; however, a core stable outcome may fail to exist (Bogomolnaia and Jackson, 2002), and deciding the existence of a core stable outcome is NP-complete (Ballester, 2004).

There are also other subclasses of hedonic games where stable outcomes are guaranteed to exist, such as acyclic hedonic games (Demange, 2004; Igarashi and Elkind, 2016), dichotomous games (Peters, 2016), and top-responsive games (Alcade and Revilla, 2004).

2. Our Model

For every positive integer , we denote by the set . We start by defining the class of games that we are going to consider.

Definition 2.1 ().

A diversity game is a triple , where and are disjoint sets of agents and for each agent it holds that is a linear order over the set

We set ; the agents in are called the red agents, and the agents in are called the blue agents.

We refer to subsets of as coalitions. For each , we denote by the set of coalitions containing . For each coalition , we say that is mixed if it contains both red and blue agents; a mixed coalition is called a mixed pair if .

For each agent , we interpret the order as her preferences over the fraction of the red agents in a coalition; for instance, if , this means that agent prefers a coalition in which two thirds of the agents are red to a coalition in which three fifths of the agents are red.

For each coalition , we denote by the fraction of the red agents in , i.e., ; we refer to this fraction as the red ratio of . For each and , we say that agent strictly prefers to if , and we say that weakly prefers to if or . A coalition is said to be individually rational if every agent in weakly prefers to .

An outcome of a diversity game is a partition of agents in into disjoint coalitions. Given a partition of and an agent , we write to denote the unique coalition in that contains . A partition of is said to be individually rational if all coalitions in are individually rational.

The core is the set of partitions that are resistant to group deviations. Formally, we say that a coalition blocks a partition of if every agent strictly prefers to her own coalition . A partition of is said to be core stable, or in the core, if no coalition blocks .

We also consider outcomes that are immune to individual deviations. Consider an agent and a pair of coalitions and . An agent accepts a deviation of to if weakly prefers to . A deviation of from to is said to be an NS-deviation if prefers to ; and an IS-deviation if it is an NS-deviation and all agents in accept it. A partition is called Nash stable (NS) (respectively, individually stable (IS)) if no agent has an NS-deviation (respectively, an IS-deviation) from to another coalition or to .

We say that the preferences of an agent are single-peaked if for every there is a peak such that

In particular, if an agent has a strong preference for being in the majority, then her preferences are single-peaked, as illustrated in the following example.

Example 2.2 (Birds of a feather flock together).

Suppose that all agents in are smokers and all agents in are non-smokers. Then we expect an agent to prefer groups with the maximum possible ratio of agents of her own type. Formally, for each and each we have if and only if , and for each and each we have if and only if . In this case, the partition in which each agent forms a singleton coalition is core stable and Nash stable (and hence also individually stable).

If an agent strongly prefers to be surrounded by agents of the other type, her preferences are single-peaked, too.

Example 2.3 (Bakers and Millers (Aziz et al., 2014)).

Suppose that each agent prefers the fraction of agents of the other type to be as high as possible. This holds, for instance, when individuals of the same type compete to trade with individuals of the other type. A Bakers and Millers game is a diversity game where for each and each we have if and only if , and for each and each we have if and only if . Note that if , a partition into mixed pairs is core stable and Nash stable; indeed, Aziz et al. (2014) prove that every Bakers and Millers game has a non-empty core.

3. Core stability

Examples 2.2 and 2.3 illustrate that if all agents have extreme homophilic or extreme heterophilic preferences, the core is guaranteed to be non-empty. However, we will now show that in the intermediate case the core may be empty, even if all agents have single-peaked preferences.

Example 3.1 ().

Consider a diversity game , where the set of agents is given by and . Agents can be divided into the following three categories with essentially the same preferences: , , and . We have , , , , , , , , , , . Each agent has the following single-peaked preferences over the ratios of red agents:

Figure illustrates the preferences of each preference category of red agents.

Figure 1. Single-peaked preferences over the ratios of red agents. The thick, grey, and dotted lines represent the preferences of agents in , , and , respectively. We omit the ratios since there is no individually rational coalition with these fractions of red agents.

We will now argue that the game in Example 3.1 has empty core.

Proposition 3.2 ().

The game in Example 3.1 has no core stable outcomes.

Proof.

Suppose towards a contradiction that there exists a core stable outcome . We note that, by individual rationality, for every mixed coalition , as red agents in a coalition with would strictly prefer to be alone. Also, contains at least one mixed coalition, as otherwise would block .

Let be the largest ratio of red agents in a mixed coalition in . We have argued that . Also, if , then for all and or ; thus coalition with blocks . Hence, . Now since contains at least one mixed coalition with red ratio at least , and all mixed coalitions in must have red ratio at least , if there is more than one mixed coalition, the number of agents would be at least 10, a contradiction. Thus, contains exactly one mixed coalition of red ratio at least . It follows that for each agent it holds that either

  • belongs to a mixed coalition of red ratio at least , i.e., ; or

  • belongs to a completely homogeneous coalition, i.e., if , then and if , then .

In particular, this means that for all , and hence each prefers to .

Now suppose that does not belong to a coalition with his favorite red ratio, i.e., . Then and thus prefers to . Thus, the coalition of red ratio blocks , a contradiction. Hence, belongs to a coalition of his favorite red ratio , and thus . Further, if some agent does not belong to a mixed coalition, then and the coalition would block . Hence, the unique mixed coalition of red ratio contains both and all four red agents in , which means that no red agent in belongs to the mixed coalition. Now we have:

  • prefers to since or ; and

  • each prefers to ; and

  • each prefers to .

Then coalition of red ratio would block , a contradiction. We conclude that does not admit a core stable partition. ∎

Indeed, we can show that checking whether a given diversity game has a non-empty core is NP-complete.

Theorem 3.3 ().

The problem of checking whether a diversity game has a non-empty core is NP-complete.

Proof.

Checking core stability can be done in polynomial time and hence our problem is in NP. Indeed, for each size of a coalition and each number , we can check in polynomial time if there is a blocking coalition of size including red players: We consider the set of all players who strictly prefer the ratio to their ratio under and verify whether admits a subset of size at least that contains red players; if this is the case, there is a coalition of size exactly that strongly blocks . If no such coalition exists, is core stable.

We reduce from the problem of deciding whether the core of an anonymous hedonic game is non-empty. A hedonic game is called anonymous if all players only care about the sizes of their coalition. Formally, an anonymous hedonic game is a pair , where is a set of agents and for each agent it holds that is a linear order over the set  of coalition sizes. Ballester (Ballester, 2004) has shown that deciding whether the core of an anonymous hedonic game is non-empty is NP-hard.

Let  be the number of players and let the preferences of the players be encoded as follows

where denotes th most preferred the coalition size of agent .111For technical reasons, we assume that always all sizes between and are listed, including irrational ones. We construct our equivalent hedonic diversity game instance as follows. First, for each original player we introduce one blue agent

Naturally, the preferences of the original player  with respect to coalitions of size  shall be represented by the preference of player  with respect to coalitions with  blue players and exactly one red player. For each possible coalition size from the original game, we introduce red players  that would accept to be in a coalition with  blue player and otherwise prefer to be alone:

If we could guarantee that every mixed coalition has at most one red player, then, the correctness proof would be almost immediate. It might, however, happen that there are coalitions with more then one red player. In the following correctness proof, we will see that these coalitions are unproblematic.

Formally, we show that the core of the original anonymous hedonic game is non-empty if and only if the core of the newly constructed hedonic diversity game is non-empty.

For the first direction, assume that the core of the anonymous game was non-empty. That means there is a partition , , , of the original players into coalitions so that no blocking coalition exists. We build a core-stable partition , , , , of our newly constructed hedonic diversity game instance from  as follows: , where and and simply contains all remaining red players.

Assume towards a contradiction that there was a blocking coalition  of players that blocks . We first argue that we can assume w.l.g. that  contains at most one red player. Of course, cannot contain only red players because every red player in is either in and, thus, already only with other red players or it is by definition of  in one of its most preferred coalition. Now, assume  contains  red players. By the definition of the preferences of the red players, must be the red player’s most preferred coalition ratio. That is, the number of blue players in  must be divisible by . This immediately implies that there is a smaller blocking coalition  which consists of one aribitrary red player from  and arbitrary blue players from . Finally, (now containing exactly one red player) can only be blocking because every blue player prefers a coalition with one red and blue players towards its current coalition in . The way we constructed the preferences of of blue players ensures that this implies that the coalition must be a blocking coalition for —a contradiction.

For the second direction, assume that the core of the diversity game was non-empty. That means there is a partition

of the blue and red players into coalitions so that no blocking coalition exists.

We assume w.l.g. that all coalitions  are mixed and  contains only red players. First, there could exist multiple purely red coalitions but merging them into one gives a partition that must be core stable if and only if the original partition was. Second, we can easily observe that is non-empty, because no player prefers mixed coalitions with a majority of red players and there are much more red than blue players. Third, a purely blue coalition cannot exist. Every blue player prefers coalitions with as many red as blue players to those coalitions and there are always enough red players available that would be willing to pair up. (Since one of the equals , ratio is preferred to .) Finally, for each integer coalition must contain at least one red player  which would prefer to be in a coalition with  blue players. Let us refer to this number as .

Furthermore, we can assume that w.l.g. each contains at exactly one red player. Every red player that is contained in some  must be in a coalition that has the player’s most preferred ratio of red player, otherwise the player would be in . In particular, this means that if there are  red players in , then all these players must be of the same type and prefer coalitions with a ratio of red players for some integer . Clearly, splitting up   into  coalitions with exactly one red player and exactly  blue players will give us another core stable partition, because these new coaltions have exactly the same ratio as  has.

With all these assumptions, we are now ready to define a partition  for the players of the original anonymous hedonic game instance via . Finally, assume towards a contradiction that there would be a coalition  that blocks . Then, using our assumptions on and the definition of the players preferences, the coalition , where and , would be a blocking coalition for : Player  comes from and, hence, clearly prefers  and the blue players, by definition of their preferences, prefer  if and only if the corresponding original players prefer a size  coalition towards their current ones—a contradiction that there is a blocking coalition for . ∎

In Example 3.1 there are at least two agents of each type. In contrast, if one of the types is represented by a single agent, then the core is guaranteed to be non-empty.

Proposition 3.4 ().

Let be a diversity game with or . Then the core of is non-empty, and a partition in the core can be constructed in polynomial time.

Proof.

Without loss of generality, assume that . Let be the set of ratios of red agents in individually rational coalitions to which belongs, i.e.,

Take ’s favorite ratio among the fractions in . Construct a coalition with and put the rest of blue agents into singletons. Clearly, the resulting partition is not blocked by any coalition of blue agents. Also, if there exists a coalition that contains the red agent and blocks , then and is individually rational, contradicting the choice of . Thus, the resulting partition is core stable. ∎

We also note that in the game in Example 3.1 agents’ preferences belong to one of the three categories. The next proposition shows that if all agents have the same preferences, then there is a core stable outcome. We conjecture that with only two types of single-peaked preferences, the core is non-empty as well.

Proposition 3.5 ().

Let be a diversity game such that each agent has the same preference over . Then has a non-empty core and a partition in the core can be constructed in polynomial time.

Proof.

We claim that our game satisfies the top coalition property: for each , there exists a top coalition such that every agent in weakly prefers to every other coalition contained in . Indeed, take any . If there is an agent who weakly prefers his own coalition to every mixed coalition contained in , then taking certifies the existence of a top coalition. Thus, suppose otherwise, i.e., every agent in strictly prefers some mixed coalition to his own singleton. As every agent has the same ranking over the fractions of red agents in mixed coalitions, there is a mixed coalition such that is the favorite red ratio for every agent in among the ratios of red agents in the subsets of , which means that is a top coalition. This argument also gives us an efficient algorithm for identifying a top coalition: it suffices to check whether some singleton forms a top coalition, and, if not, find the agents’ most preferred ratio that can be implemented in and construct a coalition with this red ratio.

We can now construct a stable partition by repeatedly identifying a top coalition with respect to the current set of agents, adding it to the partition, and removing the agents in that coalition from the current set of agents; it is clear that this procedure can be executed in polynomial time, and Banerjee et al. (2001) argue that it produces a core stable outcome. ∎

4. Nash Stability and Individual Stability

We have seen that core stability may be impossible to achieve. It is therefore natural to ask whether every diversity game has an outcome that is immune to individual deviations. It is easy to see that the answer is ‘no’ if we consider NS-deviations, even if we restrict ourselves to single-peaked preferences: for instance, the game where there is one red agent who prefers to be alone and one blue agent who prefers to be in a mixed coalition has no Nash stable outcomes.

In contrast, each diversity game with single-peaked preferences admits an individually stable outcome. Moreover, such an outcome can be computed in polynomial time. In the remainder of this section, we will present an algorithm that achieves this, and prove that it is correct. The algorithm will be divided into three parts:

  1. For agents with peaks greater than half, make mixed coalitions with red majority. For agents with peaks smaller than half, make mixed coalitions with blue majority.

  2. Make pairs from the remaining red agents and blue agents who are not in the mixed coalitions.

  3. Put all the remaining agents into singletons.

We will first show that one can construct a sequence of mixed coalitions with red majority that are immune to IS-deviations.

4.1. Create mixed coalitions with red majority

In making mixed coalitions, we will employ a technique that is similar to the algorithm for anonymous games proposed by Bogomolnaia and Jackson (2002). Intuitively, imagine that red agents and blue agents with peaks at least form two lines, each of which is ordered from the highest peak to the lowest peak. The agents enter a room in this order, with a single blue agent entering first and red agents successively joining it as long as the fraction of red agents does not exceed the minimum peak of the agents already in the room. Once the fraction of the red agents reaches the minimum peak, a red agent who enters the room may deviate to a coalition that has been formed before. We alternate these two procedures as long as there is a red agent who can be added without exceeding the minimum peak. If no red agent can enter a room, then agents start entering another room and create a new mixed coalition in the same way. The algorithm terminates if either all red agents or all blue agents with their peaks at least half join a mixed coalition. Figure 2 illustrates this coalition formation process. We formalize this idea in Algorithm 1. We will create mixed coalitions containing exactly one blue agent, so we define the virtual peak to be the favorite ratio of agent among the ratios of red agents in coalitions containing exactly one blue agent.

Figure 2. Illustration of how HALF creates coalitions with red majority. Gray circles correspond to red agents, while white circles correspond to blue agents.
input : A single-peaked diversity game
output : 
1 sort red and blue agents so that and ;
2 initialize , and ;
3 initialize and ;
4 while  and  do
5       set ;
6       while , or there exist a agent and such that has an IS-deviation from to  do
7             // add red agents to as long as the ratio of red agents does not exceed the minimum virtual peak;
8             while  do
9                   set ;
10                   set and ;
11                  
12            // let red agents in deviate to smaller-indexed coalitions;
13             if there exists an agent and such that has an IS-deviation from to  then
14                   choose and so that is ’s most preferred ratio among the coalitions satisfying the above;
15                   set , ;
16                  
17            
18      set , and ;
19      
20// let the remaining agents deviate to mixed coalitions as long as they prefer the ratio of the deviating coalition to half;
21 while there is an agent and a coalition such that , , and all agents in accept a deviation of to  do
22       choose and so that is ’s most preferred ratio among the coalitions satisfying the above;
23       and ;
24      
25if  and consists of a single blue agent  then
26       return ;
27      
28else
29       return ;
30      
Algorithm 1 HALF

In what follows, we assume that are the final coalitions that have been obtained at the termination of the algorithm, and that is the output returned by the algorithm. For each and ,

  • is called a default agent of if is a blue agent or is a red agent who has joined at Step 1;

  • is called a new agent of if is a red agent who has joined in Step 1 or Step 1.

We denote by the set of default agents of , and we denote by the set of new agents of .

For each with , each agent in is either a default agent of a new agent. Notice that starts with the empty set and plays the role of the last resort option for red agents, i.e., red agents can always deviate to if they strictly prefer staying alone to the mixed coalition they have joined.

Deviations of blue agents  We will first show that the algorithm constructs a sequence of coalitions such that no blue agent in the coalitions has an IS-deviation to other coalitions. We establish this by proving a sequence of claims. First, it is immediate that the red ratio of each coalition is at least half, except for the last coalition, which may contain a single blue agent.

Lemma 4.1 ().

For each , the red ratio of is at least half, i.e., .

Proof.

Take any . The claim is immediate when . Suppose that for some . Then it is clear that contains exactly one blue agent. If contains no red agent, then this would mean that all red agents with their peaks at least half joined , but deviated to smaller indexed-coalitions, meaning that is the last coalition that has been formed, i.e., ; but this contradicts the construction in Step 1. Thus, contains exactly one blue agent and at least one red agent, which implies that . ∎

This leads to the following lemma.

Lemma 4.2 ().

For each , is a red agent and strictly prefers to a mixed pair.

Proof.

By construction every agent in is a red agent. Suppose that before joining , has joined a mixed coalition with at Step 1 and then deviated to at Step 1. Just before has left , the ratio of red agents in is at least half, implying that or . Since joined at Step 1, we also have . Combining these yields . The claim is immediate when joined at Step 1. ∎

We also observe that, by the construction of the algorithm, all default agents belong to the coalition whose red ratio is at most their virtual peak; see Figure (2(a)) for an illustration.

Lemma 4.3 ().

For every default agent in , where with ,

  1. the red ratio of is at most ’s virtual peak, i.e., ; and

  2. weakly prefers to and to a mixed pair.

Proof.

Take any and take any default agent in . Before the algorithm starts forming the next coalition , the ratio remains below by the while-condition in Step 1. After starts being formed, can only increase by accepting red agents from larger-indexed coalitions. However, if a deviation of some red agent to increases the fraction above , this would mean that would not be willing to accept such a deviation. Thus, . To show the second statement, recall that by construction of the algorithm and by Lemma 4.1. If is a blue agent, then by single-peakedness weakly prefers both to her own singleton and to a mixed pair. If is a red agent who has joined at Step 1, then by single-peakedness and the fact that he has not deviated to at Step 1, weakly prefers both to his singleton and to a mixed pair. ∎

We are now ready to prove that no coalition in admits a deviation of a blue agent.

Lemma 4.4 ().

For each , there is an agent in who does not accept a deviation of a blue agent to .

Proof.

If and , it follows from Lemma 4.2 that no accepts a deviation of a blue agent. Suppose for some , and assume towards a contradiction that some blue agent can be accepted by all agents in for some . By Lemma 4.1, the ratio of red agents in is at least . If the fraction of red agents in is at most the peak of some agent, i.e., for some agent , this means that

i.e., would not accept , a contradiction. Hence, suppose that the fraction of red agents in is beyond the maximum peak, i.e., . Take . By Lemma 4.3,

Since by construction of the algorithm, we have , which means that coalition contains one blue agent and at least two red agents. Hence, even if one red agent leaves the coalition, its red ratio would be at least half, i.e.,

where the second inequality holds, since otherwise , which contradicts the fact that is ’s favorite ratio of the coalitions containing exactly one blue agent. Further, agent prefers to by single-peakedness and by the fact that . Since accepts the deviation of to , after adding to , the red ratio should remain at least , implying that . But this would mean that , or, equivalently, . This inequality can be simplified to , implying , a contradiction. ∎

(a) Virtual peak of a default agent.

(b) Virtual peak of a new agent.
Figure 3. The red ratio of a coalition does not exceed the virtual peak of a default agent, but exceeds the virtual peak of a new agent.

Deviations of red agents  We will now show that coalitions do not admit an IS-deviation by red agents. To this end, we first observe that the red ratio of a coalition to which a new agent belongs exceeds the virtual peak of the new agent; see Figure (2(b)) for an illustration. Due to single-peakedness, this means that new agents do not accept further deviations of red agents.

Lemma 4.5 ().

Let be the set of agents that belong to with after the while-loop of Step 1 of Algorithm 1, and let be the last agent in to join . Then no red agent with can join without exceeding the minimum peak, i.e., .

Proof.

Take any with . If , then this would mean that , since . Hence the algorithm would have added to , a contradiction. ∎

Lemma 4.6 ().

For all with and each new agent :

  1. the red ratio of exceeds ’s virtual peak, i.e., ;

  2. is the unique new agent in ;

  3. weakly prefers to and to a mixed pair.

Proof.

Let be the first new agent who joined at Step 1 or Step 1. When joined , his virtual peak was at most that of any other red agent in . By Lemma 4.5, he cannot join without exceeding the virtual peak , i.e., . Thus, by single-peakedness, he does not accept any further deviations by red agents. Hence, is the unique new agent in . To see that the third statement holds, note that if joined at Step 1, then prefers to by the if-condition of Step 1. Also, chooses his favorite coalition among the coalitions to which he can deviate, and thus prefers to . Now suppose that joined from with at Step 1. At that point, the ratio of red agents in was at least half, and agent weakly prefers to a mixed pair since . Hence, since prefers to , by transitivity prefers to . Also, by the if-condition in Step 1, prefers to being in a singleton coalition. ∎

We also observe that the red ratio of a higher-indexed coalition is smaller than or equal to that of a lower-indexed coalition.

Lemma 4.7 ().

Let and let be the set of agents in just after the while-loop of Step 1. Then ; in particular, .

Proof.

Assume for the sake of contradiction that . Since and contain exactly one blue agent, this means that contains more red agents than does, i.e., . Thus, even if we add a red agent to , its red ratio does not exceed , i.e., for each we have . Now recall that by Lemma 4.3, the red ratio of does not exceed the virtual peak of each default agent, which means that

Combining these observations, for every we have

(4.1)

Moreover, since , or has been created after , there is a red agent with for every and , where and are the unique blue agents in and , respectively. Combining this with inequality (4.1) yields

which contradicts Lemma 4.5. ∎

Lemma 4.8 ().

Let with . Then some agent in does not accept a deviation of a red agent to .

Proof.

Since , both and contain exactly one blue agent as well as the same number of red agents. By the description of the algorithm, the minimum virtual peak of agents in is smaller than that of agents in , i.e.,

(4.2)

Further, by Lemma 4.5, no red agent in can join without exceeding the minimum virtual peak, which implies that for every we have

Combining this with the inequality (4.2) implies

Thus, by single-peakedness, there is an agent in who does not accept a deviation of a red agent to . ∎

We are now ready to show that no coalition in admits a deviation of a red agent.

Lemma 4.9 ().

No agent has an IS-deviation to another coalition with .

Proof.

Suppose towards a contradiction that there is such an agent and a coalition with . Let be the largest index of coalition to which has an IS-deviation. Observe that if contains a new agent, then the new agent does not accept a deviation of agent by Lemma 4.6; thus, has no new agent and . If is a default agent in , then weakly prefers to and hence strictly prefers to by transitivity. Thus, agent could have deviated to