On Colorful Bin Packing Games

11/09/2017 ∙ by Vittorio Bilò, et al. ∙ Università degli Studi dell'Aquila GranSassoScienceInst 0

We consider colorful bin packing games in which selfish players control a set of items which are to be packed into a minimum number of unit capacity bins. Each item has one of m≥ 2 colors and cannot be packed next to an item of the same color. All bins have the same unitary cost which is shared among the items it contains, so that players are interested in selecting a bin of minimum shared cost. We adopt two standard cost sharing functions: the egalitarian cost function which equally shares the cost of a bin among the items it contains, and the proportional cost function which shares the cost of a bin among the items it contains proportionally to their sizes. Although, under both cost functions, colorful bin packing games do not converge in general to a (pure) Nash equilibrium, we show that Nash equilibria are guaranteed to exist and we design an algorithm for computing a Nash equilibrium whose running time is polynomial under the egalitarian cost function and pseudo-polynomial for a constant number of colors under the proportional one. We also provide a complete characterization of the efficiency of Nash equilibria under both cost functions for general games, by showing that the prices of anarchy and stability are unbounded when m≥ 3 while they are equal to 3 for black and white games, where m=2. We finally focus on games with uniform sizes (i.e., all items have the same size) for which the two cost functions coincide. We show again a tight characterization of the efficiency of Nash equilibria and design an algorithm which returns Nash equilibria with best achievable performance.

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I Introduction

A classical problem in combinatorial optimization is the one-dimensional bin packing problem, in which items with different sizes in

have to be packed into the smallest possible number of unit capacity bins. This problem is known to be NP-hard (see [8] for a survey).

The study of bin packing in a game theoretical context has been introduced in [4]. In such a setting, items are handled by selfish players and the unitary cost of each bin is shared among the items it contains. In the literature, two natural cost sharing functions have been considered. The egalitarian cost function, used in [9, 18], which equally shares the cost of a bin among the items it contains, and the proportional cost function, studied in [4, 10], where the cost of a bin is split among the items proportionally to their sizes. That is, each player is charged with a cost according to the fraction of the used bin space her item requires. We notice that for games with uniform sizes, i.e., when all the items have the same size , the two cost functions coincide. Each player would prefer to choose a strategy that minimizes her own cost, where the strategy is the bin chosen by the player. Pure Nash equilibria, i.e. packings in which no player can lower her cost by changing the selected bin in favor of a different one, are mainly considered as natural stable outcomes for these games. The social cost function that we aim to minimize is the number of open bins (a bin is open if it stores at least one item). Bin packing games can be used to model many practical scenarios, like for instance, bandwidth allocations problems, packet scheduling problems, etc. (see [4, 13]).

In this paper we consider colorful bin packing games, where we are given a set of selfish players, a set of colors, and a set of unit capacity bins. Players control indivisible colored items of size in . Each item needs to be packed into a bin without exceeding its capacity and in such a way that no item is misplaced, that is no item is packed next to an item of the same color. The special case of games with two colors is called black and white bin packing games. We use both the egalitarian and the proportional cost functions, where we set the cost of any misplaced item as infinite, and adopt Nash equilibria as stable outcomes of the games. Clearly, in any Nash equilibrium no player can be charged with a infinite cost, since, for any player, moving a misplaced item to an empty bin (and thus getting cost ) is always an improving deviation. We notice that, if all the items have different colors, these games correspond to the bin packing ones. However, when there are items with the same color, the stable outcomes of the games are structurally different than bin packing ones. In fact, we show that in our games, Nash equilibria perform very differently than in bin packing ones.

Colorful bin packing games generalize the bin packing games and therefore model many practical scenarios and situations where it is important distinguishing or separating the items, that are not captured by bin packing games. For instance, suppose that a shipping company rents out its means of transport like trucks. Customers want to ship their items at the smallest cost. For example, items could be chemical agents with different characteristics or properties such that, if two items with the same property are packed next to the other in the truck, they could dangerously react. Colorful bin packing games can be used to model such scenario. Items here correspond to chemical agents, colors to characteristics, and bins to trucks. Other interesting applications can be found in [6, 11].

Our Contribution. We first focus on the existence of Nash equilibria. We show that colorful bin packing games may not converge to Nash equilibria. In fact, they may admit an infinite sequence of improving deviations (i.e., the finite improvement path property is not guaranteed) even for special cases in which games have only two colors and uniform sizes (Proposition III.1). In this case the egalitarian and proportional cost functions coincide. However, in Theorems III.2 and III.3, we show that, under both cost functions, if one allows the players to perform only improving deviations towards bins in which no item is misplaced, then any game possesses the finite improving path property. As in any non-equilibrium profile there always exists a player who possesses one such a deviation, and in particular a misplaced item can always move to an empty bin, it follows that Nash equilibria are guaranteed to exist under both cost functions. We also show a very natural and simple algorithm Algorithm 1 (a similar approach was already considered in [10]), that computes a Nash equilibrium whose running time is polynomial under the egalitarian cost function and pseudo-polynomial for a constant number of colors under the proportional one (Theorem III.9).

We then measure the quality of Nash equilibria using the standard notions of (price of anarchy) and (price of stability), that are defined as the worst/best case ratio between the social cost of a Nash equilibrium and the cost of a social optimum, which corresponds to the minimum number of open bin needed to feasibly pack all colored items. We provide a complete characterization of the efficiency of Nash equilibria by showing that, under both cost functions, the prices of anarchy and stability of colorful bin packing games are unbounded (we consider the absolute approximation ratio), when (Theorems III.4 and III.5), while they are equal to when (Theorems IV.2, IV.3, and IV.9). We also consider the basic setting in which all items have the same size and again provide a complete picture of the efficiency of Nash equilibria which happens to depend on the parity of the number of items that can be packed into a bin without exceeding its capacity (in this case the egalitarian and proportional cost functions coincide). In particular, we show that, when is even, the price of stability is for any (Theorems V.1 and V.2), while the price of anarchy is for (Theorem V.7), and unbounded for (Theorem V.5). When

is odd, the price of stability is

for any (Theorem V.2), while the price of anarchy is for (Theorems IV.2 and V.6), and unbounded for (Theorem V.5). We also design an algorithm (Algorithm 2) which returns a Nash equilibria which is socially optimal when is odd and -approximates the social optimal when is even.

Due to space constraints, some proofs have been removed. All the details will appear in the full version of the paper.

Related Work. The classical one-dimensional bin packing problem has been widely studied (see [8] for a general survey). Bin packing games under the proportional cost function have been introduced in [4]. The author proved the existence of Nash equilibria by showing that the best-response dynamics converge in finite time. He also established that there is always a Nash equilibrium with minimal number of bins, i.e., the is 1, but that finding such a good equilibrium is NP-hard. Finally, he presented upper and lower bounds on the . Nearly tight bounds on the have been later shown in [13]. Yu and Zhang [19] have designed a polynomial time algorithm which returns a Nash equilibrium. Bin packing games under the egalitarian cost function were considered in [18]. They showed tight bounds on the and the and design a polynomial time algorithm for computing a Nash equilibrium. In [9], the authors provided tight bounds on the exact worst-case number of steps needed to reach a Nash equilibrium. Other types of equilibria (like for instance strong equilibria) and other bin packing games were also considered in [1, 7, 10, 12, 13, 14, 16].

The offline version of the black and white bin packing problem was considered in [3]. Most of the literature on colorful bin packing is about the online version of the problem. Competitive algorithms for the online colorful bin packing problem were presented in [11]. The special case black and white was considered in [2], while, the one where all items have size , was considered in [5]. All such results on the online version of the problem were improved in [6].

Related colorful bin packing problems have been also considered. For instance, in the bin coloring [17], the problem is to pack colored items into bins, such that the maximum number of different colors per bin is minimized. The bin coloring games were considered in [15], where players control colored items and each player aims at packing its item into a bin with as few different colors as possible.

To the best of our knowledge, this is the first paper dealing with Nash equilibria in colorful bin packing games.

Ii Model and Preliminaries

In a colorful bin packing game we have a set of players , a set of colors and a set of unit capacity bins . Each player controls an indivisible item, denoted for convenience as (i.e., we denote by the set of items), having size and color which needs to be packed into one bin in without exceeding its capacity. Game has uniform sizes if for every . The special case in which has colors is called the black and white bin packing game; we shall define color as black, color as white and denote by and the number of black and white items in , respectively.

A strategy profile is modeled by an -tuple such that, for each , is the bin chosen by player . We denote by the set of items packed into according to the strategy profile . Similarly, we also write to indicate the set of items packed in the same bin as (i.e., the bin chosen by player ), according to . Given any bin , we assume to pack the items in a fixed internal order, going from bottom to top, that is the sequential order in which players have chosen the bin as strategy. Namely, for any pair of items and in , we say that precedes inside the bin , and we write , if player chose bin before . Formally, given any strategy profile , each item occupies a precise position in the sequential order of items in bin , counting from bottom to top, computed as . For convenience, if the strategy profile is clear from the context we may omit the symbol and simply write . We notice that, with such packing, the last player, say , choosing the bin , occupies the top position in .

Denoted by the total size of items packed into , we always assume that , so that every strategy profile induces a packing of items in and vice versa.

We say that an item is misplaced if there exists an item with such that and , that is, is packed next to an item of the same color. A bin is feasible if it stores no misplaced items. In particular, an empty bin is feasible. A strategy profile is feasible if so are all of its bins. For games with uniform sizes for every , we denote by the maximum number of items that can be packed into any (even non-feasible) bin. We only consider the cases in which as, otherwise, the game is trivial.

We shall denote by the cost that player pays in the strategy profile and each player aims at minimizing it. We consider two different cost functions: the egalitarian cost function and the proportional cost function. We have under both cost functions when is a misplaced item, while, for non-misplaced ones, we have under the egalitarian cost function and under the proportional one. Note that, for games with uniform sizes, the two cost functions coincide. For a fixed strategy profile , we say that a bin is a singleton bin if it stores only one item. Moreover, when considering the egalitarian (resp. proportional) cost function, we denote by the bin storing the maximum number of items (resp. the fullest bin) in the packing corresponding to , breaking ties arbitrarily.

A deviation for a player in a strategy profile is the action of changing the selected bin in favor of another bin, say , such that . We shall denote as the strategy profile realized after the deviation. Formally, is defined as follows: and for each player .

In this paper, we consider deviations of the following form: is removed from and packed on top of , consistently with the sequential order of items in a bin.

An improving deviation for a player in a strategy profile is a deviation towards a bin such that . Fix a feasible strategy profile . Under the egalitarian cost function, player admits an improving deviation in if there exists a bin such that the item on top of has a color different than and . Under the proportional cost function, player admits an improving deviation in if there exists a bin such that the item on top of has a color different than and . Conversely, when a strategy profile is unfeasible, under both cost functions, a player controlling a misplaced item always possesses an improving deviation, for instance, by moving to an empty bin which is always guaranteed to exist as there are items, bins and the bin storing is non-singleton. We note that, as a side-effect of an improving deviation, may be unfeasible even if is feasible: this happens when separates two items of the same color. We say that an improving deviation is valid whenever the destination bin is feasible before the deviation.

A strategy profile is a (pure) Nash equilibrium if for each and , that is, no player has an improving deviation in . Let denote the set of Nash equilibria of game . It is easy to see that any Nash equilibrium is a feasible strategy profile. This implies that would force each item to be packed into a different bin: this justifies our choice of . A game has the finite improvement path property if it does not admit an infinite sequence of improving deviations. Clearly, if enjoys the finite improvement path property, it follows that .

Given a strategy profile , let be the set of open bins in , where a bin is open if it stores at least one item. Let be the number of open bins in , i.e., . We shall denote with the social optimum, that is, any strategy profile minimizing function F. It is easy to see that any social optimum is a feasible strategy profile.

The price of anarchy of is defined as , while the price of stability of is defined as . Given a class of colorful bin packing games , the prices of anarchy and stability of are defined as and . Let denote the set of all colorful bin packing games with colors and (resp. ) denote the set of all colorful bin packing games with colors and uniform sizes for which is odd (resp. even). Finally, denote .

Iii Existence and Efficiency of Nash Equilibria in General Games

In this section, we first show that, without any particular restriction on the type of improving deviations performed by the players, even games with uniform sizes and only two colors may not admit the finite improvement path property (Proposition III.1). However, if one allows the players to perform only valid improving deviations, then any game possesses the finite improving path property under both cost functions (Theorems III.2 and III.3). These two theorems, together with the fact that in any strategy profile which is not a Nash equilibrium there always exists a valid improving deviation, imply the existence of Nash equilibria for colorful bin packing games under both cost functions.

Proposition III.1

There exists a black and white bin packing game with uniform sizes not possessing the finite improvement path property.

Proof:

Let be a black and white bin packing game with uniform sizes defined by three black items, denoted as , and , and three white items, denoted as , and . All items have size , so that . Figure 1 depicts a cyclic sequence of (non-valid) improving deviations which shows the claim.

Fig. 1: A cyclic sequence of (non-valid) improving deviations.
Theorem III.2

If players are restricted to perform only valid improving deviations, then each colorful bin packing game under the egalitarian cost function admits the finite improvement path property.

Proof:

To prove the claim, we define a suitable potential function which strictly increases each time a player performs a valid improving deviation. Given a strategy profile , consider the potential function

and assume that a player performs a valid improving deviation by moving onto bin . We distinguish between two cases.

If bin is not feasible since both and are feasible, we obtain when is open and otherwise.

If is feasible, under the egalitarian cost function, this implies that . Observe that, since is feasible, , which implies that it must be . For the ease of notation, set and , so that and ; we get

where the second inequality comes from and the third one comes from . Thus, in any case, which, since the number of possible strategy profiles is finite, implies the claim.

The next proof uses a different function than the one used above.

Theorem III.3

If players are restricted to perform only valid improving deviations, then each colorful bin packing game under the proportional cost function admits the finite improvement path property.

Proof:

To prove the claim, we define a suitable potential function which strictly increases each time a player performs a valid improving deviation. Denote as the power set of and, given a set , denote . Define

and let be a number such that . Given a strategy profile , consider the potential function

and assume that a player performs a valid improving deviation by moving onto bin , which implies . We distinguish between two cases.

If bin is not feasible, since both and are feasible, we obtain .

If is feasible, under the proportional cost function, this implies that . For the ease of notation, set and , so that , which, by the definition of , implies that ; we get

where the last inequality comes from the definition of . Thus, in any case, which, since the number of possible strategy profiles is finite, implies the claim.

In the following we give a tight characterization of the efficiency of Nash equilibria in colorful bin packing games under both cost functions. This is achieved by giving upper bounds on the and matching or asymptotically matching lower bounds on the . For games with at least three colors, Theorems III.4 and III.5 show that, under both cost functions, the can be unbounded, thus, in the worst-case, no efficient Nash equilibria are guaranteed to exist.

Theorem III.4

Under the egalitarian cost function, is unbounded for each .

Proof:

Fix a value . We show the theorem by proving that there exists a game with players such that

By taking the limit for going to , the theorem will follow.

We construct game as follows. There are:

  • white items of size ,

  • for every color other than white, items of size ,

where and are arbitrary integers such that and is a multiple of , while is an arbitrarily small real value such that .

Observe that, since a combination of white items and non-white items can be feasibly packed into the same bin, and there are exactly white items and non-white items, there exists a feasible strategy profile using exactly bins. Moreover, note that white items do not fit into a bin, as their total size is equal to by the definition of .

We proceed by showing that any Nash equilibrium for needs to use at least bins. To this aim, fix a Nash equilibrium for and consider bin .

If the item on top of is white, then all non-white items have to be stored in . In fact, since can contain at most white items, for an overall occupation of , it follows that has enough space to accommodate all the non-white items. Thus, since stores all the non-white items and can store at most white items, it follows that needs to use at least singleton bins to feasibly store all white items not stored in , so that .

Hence, assume that the item of top of has a color which is not white. By the same arguments exploited above, it follows that have to contain all non-white items having color different than . Again, since can contain at most white items, needs to pack at least white items outside bin . To do this, at most non-white items (the ones having color ) can be used. The only way to pack these items so that results in a Nash equilibrium is to use a first-fit-like algorithm. Note that, in order to pack white items in the same bin, at least item of color are needed. Hence, by using items of color , at most white items can be packed in a non-singleton bin, so that at least white items need to be stored in a singleton bin.

Thus, we can conclude that . Set for , for , and for . For sufficiently high values of , by using , the claim follows.

Theorem III.5

Under the proportional cost function, is unbounded for each .

Proof:

We prove the claim under the hypothesis of . It is easy to adapt the proof so as to deal with any number of colors .

Consider the following instance with players, where is a multiple of . There are items of color , of color , and of color . The sizes are such that, one item of color has size , while all the others ones have size . Finally, each item of color or has size . The values of the sizes are as follows:

  • , for any (i.e., ).

Notice that , therefore any solution cannot use less than two bins. An optimal solution can be achieved by assigning the item of size to one bin, and all the other items to a second bin, as depicted in Figure 2. Notice that, (i) the total size of items packed in the second bin is at most . Moreover, (ii) , i.e., the item of size and all the items of size can be packed together in a bin.

We further notice that, the item of size cannot be packed with any other item of the same color , because at least one element of different color is needed between them, and .

Consider any Nash equilibrium where the item of size is packed in a bin . We now show that at most two items of size , are not packed in . Let us suppose, by contradiction, that there exist at least three items of size that are not packed in the bin . If among such three items, there exist one of color and another one of color , then, by properties (i) and (ii), we get a contradiction with the fact that it is a Nash Equilibrium. Thus, the three items must have the same color, and without loss of generality, suppose that it is . In this case we also get a contradiction. Indeed, it is easy to see that, it is not possible to pack items of color in the bin , by using at most items of color . We conclude by noticing that, at least items of sizes must be packed into singleton bins, and this concludes the proof.

Fig. 2: An optimal configuration for the instance considered in Theorem III.5.

We conclude the section by presenting a simple algorithm, namely Algorithm 1, for computing a Nash equilibrium in colorful bin packing games under both cost functions. In particular, we shall prove that its running time is polynomial for the egalitarian cost function and pseudo-polynomial for the proportional one for the special case of constant number of colors. Algorithm 1 is based on the computation of a solution for the following two optimization problems.

Max Cardinality Colorful Packing: Given a set of items , where, for each , has size and color , compute a set of items, of maximum cardinality, which can be packed into a feasible bin without exceeding its capacity.

Colorful Subset Sum: Given a set of items , where, for each , has size and color , compute a set of items, of maximum total size, which can be packed into a feasible bin without exceeding its capacity.

1:
2:while  do
3:     if ( is defined under the egalitarian cost function) then
4:         Let be a solution to Max Cardinality Colorful Packing()
5:     else
6:         Let be a solution to Colorful Subset Sum()
7:     end if
8:     Open a new bin and assign it the set of items
9:     
10:end while
11:return the strategy profile induced by the set of open bins
Algorithm 1 It takes as input a colorful bin packing game

Next lemma shows the correctness of the algorithm.

Lemma III.6

Algorithm 1 computes a Nash equilibrium for any colorful bin packing game .

Proof:

Consider the strategy profile returned by Algorithm 1 and assume, for the sake of contradiction, that it is not a Nash equilibrium. Then, there exist an item packed into a bin and a bin such that can be feasibly packed into and under the egalitarian cost function, while under the proportional one. Now, if is opened before by Algorithm 1, we have that the set of items packed into together with is a better packing than , thus contradicting its optimality. Conversely, if is opened before by Algorithm 1, we have that the set of items packed into together with is a better packing than , thus contradicting its optimality. Hence, in both cases, we get a contradiction.

We are also able to show that Max Cardinality Colorful Packing can be efficiently solved.

Lemma III.7

Max Cardinality Colorful Packing can be solved in polynomial time.

Proof:

Our proof is based on the following two fundamental observations: (i) a set of colored items can be feasibly packed into a bin without exceeding its capacity if and only if the frequency of the dominant color in is at most , (ii) if there exists a set of colored items that can be feasibly packed into a bin without exceeding its capacity whose dominant color has frequency , then the set of items computed as follows:

  • choose the items of color having minimum size;

  • let be the set of items obtained by choosing from , for each color other than , the items of minimum size (or all items if there are less than items of that color);

  • number all items in in non-decreasing order of sizes;

  • choose the first items in ;

can also be feasibly packed into a bin without exceeding its capacity.

Let be an optimal solution to the problem, be the dominant color in and be the frequency of the items of color in . If we knew both and , because of observation (ii), a solution such that can be constructed in time . Thus, the claim follows by guessing all possible pairs , with , and then returning the best feasible solution.111Indeed, it is possible to lower the computational complexity of the algorithm by considering only the pairs by noting that the algorithm used within observation (ii) can be easily adapted to deal with the case in which is not known.

For what concerns Colorful Subset Sum, the problem can be solved in pseudo-polynomial time as long as the number of colors is constant.

Lemma III.8

Colorful Subset Sum can be solved in pseudo-polynomial time as long as the number of colors is constant.

Proof:

Assume to know the exact number of items of each color that belong to an optimal solution. Note that Colorful Subset Sum becomes a variant of a Knapsack problem in which item are partitioned into sets according to their colors and one wants to maximize the sum of the volumes of the items packed in the knapsack without exceeding its capacity by choosing exactly items from each set. By suitably extending the dynamic programming algorithm for the knapsack problem and guessing the possible tuples such that , the claim follows.

As a consequence of Lemmas III.6, III.7 and III.8, we obtain the following result.

Theorem III.9

A Nash equilibrium for colorful bin packing games can be computed in polynomial time under the egalitarian cost function and in pseudo-polynomial time for a constant number of colors under the proportional one.

Iv Efficiency of Nash Equilibria in Black and White Games

For black and white bin packing games, things get much more interesting, as we show an upper bound of on the and a corresponding lower bound on the . To address this particular case, given a black and white bin packing game , we make use of the following additional notation. Given a strategy profile , we denote by the set of singleton bins storing a black item, by the set of singleton bins storing a white item, by the set of non-singleton bins having a black item on top, and by the set of non-singleton bins having a white item on top.

The following lemma relates the set of open bins of a feasible strategy profile with that of a social optimum.

Lemma IV.1

Fix a feasible strategy profile and a social optimum for a black and white bin packing game . Then, .

Proof:

First, we observe that, since in any open bin of the absolute value of the difference between the number of black and white items is at most , we have

(1)

Now, let us denote with the number of black items packed into bins belonging to and with the number of white items packed into bins belonging to . We have and . Moreover, the number of black items packed into bins belonging to is at least the number of white items packed into bins belonging to minus , so that, by putting all together, we obtain

(2)

By combining inequalities (1) and (2), the claim follows.

The following theorem gives an upper bound on the of black and white bin packing games under both cost functions.

Theorem IV.2

Under both cost functions, .

Proof:

Given a black and white bin packing game under a certain cost function, fix a Nash equilibrium and a social optimum . Let be the sum of the sizes of all the items. Notice that . Assume without loss of generality that (if this is not the case, we simply swap the two colors).

Let be the set of pairs of bins constructed as follows: each bin in is paired with a bin in , each remaining bin in is paired with a bin in , finally, all the remaining bins in and all the bins in are joined into pairs until possible. It is easy to check that, for each created pair of bins , it must be

(3)

under both cost functions, otherwise the hypothesis that is a Nash equilibrium would be contradicted. Moreover, by (3), it follows that which implies that .

Now two cases may occur:

  • no bin in is left unmatched by , which implies that , as at most one bin from the set may remain unmatched. Thus, we obtain

  • at least one bin in is unmatched by , which implies that . Thus, we obtain

    where the second inequality comes from Lemma IV.1.

In the next two theorems, we show a matching lower bound on the of black and white bin packing games under both cost functions.

Theorem IV.3

Under the egalitarian cost function, .

Proof:

We prove the theorem by showing that, for any , there exists a black and white bin packing game such that . is defined by the following set of items:

  • white items having size , denoted as items of type (1);

  • black items having size , denoted as items of type (2);

  • black items having size , denoted as items of type (3);

  • white items having size , denoted as items of type (4);

where is an even integer such that and is arbitrarily small.

Denote with the strategy profile such that with and such that

  • bin contains items of type (1), items of type (4) and items of type (3),

  • each bin from to , for a total of bins, contains one item of type (1),

  • each bin from to , for a total of bins, contains one item of type (2).

In the definition of , we avoid considering the order in which the items are packed within each bin as it is irrelevant to our purposes. We only stress the fact that there exists a proper ordering of the items which makes a feasible strategy profile.

Now, denote with the strategy profile such that with and such that

  • bins and both contain items of type (1) and items of type (3),

  • each bin from to , for a total of bins, contains one item of type (2) and 2 items of type (4).

Also in this case, we avoid considering the order in which the items are packed within each bin and stress the fact that there exists a proper ordering of the items which makes a feasible strategy profile.

In order to show the claimed lower bound on , we proceed by proving that the packing of items corresponding to any Nash equilibrium for coincides with the one corresponding to . This is achieved by exploiting a sequence of results.

First of all, we observe the following basic fact.

Fact 1

In any Nash equilibrium for , any bin containing an item of type (2) can store at most items.

We continue by proving some basic properties possessed by bin , for any Nash equilibrium for .

Lemma IV.4

Fix a Nash equilibrium for . Then, stores at least 4 items.

Proof:

Assume, for the sake of contradiction, that stores at most 3 items. Since the total number of items is , it follows that is made up of at least bins. This implies that .

Since : a contradiction to Theorem IV.2.

As a consequence of Fact 1 and Lemma IV.4, we get the following corollary.

Corollary IV.5

Fix a Nash equilibrium for . No item of type (2) is packed into .

Lemma IV.6

Fix a Nash equilibrium for . All items of type (4) are packed into .

Proof:

Assume, for the sake of contradiction, that there exists an item of type (4), say , which is not packed into . Since is a Nash equilibrium, the item on top of must be white, otherwise player would lower her cost by migrating to . By Corollary IV.5, can only store items of type (1), (3) and (4). The load coming from items of type (1) packed into can be at most (since at most items of type (1) can be packed into the same bin), so that bin can potentially store all items of type (3). This implies that all of these items must indeed be packed into , otherwise the player owning any of the leftover ones would lower her cost by migrating to . Hence, we can conclude that stores black items and the item on top of is white. This implies that has to store at least white items. Now, since at most items of type (1) can be packed into the same bin, needs to store all items of type (4).

Lemma IV.7

Fix a Nash equilibrium for . All items of type (3) are packed into .

Proof:

Assume, for the sake of contradiction, that there exists an item of type (3), say , which is not packed into . Again, by Corollary IV.5, can only store items of type (1), (3) and (4) and the load coming from items of type (1) packed into can be at most , so that bin can potentially store all items of type (3). Hence, since is a Nash equilibrium, the item on top of must be black, otherwise player would lower her cost by migrating to . This implies that the total load of has to exceed , otherwise any player owning an item of type (1) would lower her cost by migrating to . Given that no item of type (2) can be stored in , we have that, in order to achieve a load of more than , has to store exactly items of type (1). Moreover, by Lemma IV.6, all items of type (4) are packed in . Hence, we can conclude that stores exactly white items and the item on top of is black. This implies that has to store at least black items none of which belonging to type (2), that is, has to store all items of type (3).

We can finally prove that all Nash equilibria for correspond to the same packing of items.

Lemma IV.8

Fix a Nash equilibrium for . Then, and are equal up to a renumbering of the bins.

Proof:

Fix a Nash equilibrium for . By Lemmas IV.6 and IV.7, it follows that stores all items of type (3) and (4), that is, black items and white items. Hence, in order to obtain a feasible strategy profile, needs to store at least items of type (1). Moreover, cannot store more than items of type (1). If stores items of type (1), the player owning any of the leftover items of type (1) would lower her cost by migrating to . This implies that needs to store exactly items of type (1). Since the remaining items of type (1) and all items of type (2) can only be packed into different bins, it follows that there exists a suitable renumbering of the bins in which gives .

We can conclude our proof by lower bounding the price of stability of . Because of Lemma IV.8, we get

since implies that .

Theorem IV.9

Under the proportional cost function, .

Proof:

We prove the theorem by showing that, for any , there exists a black and white bin packing game such that . is defined by the following set of items:

  • white items having size , denoted as items of type (1);

  • black items having size , denoted as items of type (2);

  • black items having size , denoted as items of type (3);

  • white items having size , denoted as items of type (4);

where is an even integer such that and is an arbitrarily small number satisfying .

Denote with the strategy profile such that with and such that

  • bin contains items of type (1), items of type (4) and items of type (3),

  • each bin from to , for a total of bins, contains one item of type (1),

  • each bin from to , for a total of bins, contains one item of type (2).

In the definition of , we avoid considering the order in which the items are packed within each bin as it is irrelevant to our purposes. We only stress the fact that there exists a proper ordering of the items which makes a feasible strategy profile.

Now, denote with the strategy profile such that