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Testing and Comparing Multi-objective Evolutionary Algorithms for Multi-payoff Games

Work-in-progress

by

Eisenstadt, E., Moshaiov, A., and Avigad, G.

Note! The minimizer is 𝑷𝟏 she, and the maximizer is 𝑷𝟐 he

1 Introduction

In conventional game theory the usual assumption is that decision makers (players) make

their decisions based on a scalar payoff. But in many practical problems, in economics and

engineering, decision makers must cope with multiple objectives or payoffs. Furthermore,

each decision maker might deal with contradicting objectives. In such problems [1]–[3], a

vector of objective functions must be considered.

If players in a MOG have objective preferences then a utility function can be used to

transform the MOG into a game with a single objective. But, methods that involve a utility

function cannot be used when players do not have preferences toward objectives. Moreover,

players may want to postpone the decision to stage after all possible results and their tradeoffs

are examined. This paper deals with such situations of MOGs with no objective preferences.

In the case of zero-sum Multi-Objective Games (MOGs) one player aims at minimizing a

set of objectives, while the other aims at maximizing these objectives [1], [2], [4]. Further-

more, the gain of one player with respect to any of the objectives is the loss of the other player

with respect to that objective.

In a single objective game, when a player is said to be rational it means that it seeks to

play in a manner which maximizes its own payoff. It is also assumed that the rationality of all

players is a common knowledge so each player knows that the opponent will always choose

its most profitable strategy [5]. Moreover, it is assumed that each player thinks strategically,

meaning that it forms beliefs by analyzing what the other player might do, and then chooses a

rational response based on its beliefs [6].

The solution approach to MOGs defined here aims to support decision making i.e., sup-

port the decision maker to eventually pick a solution. It is further noted that the assumption of

a lack of information regarding the opponent's objective preferences also holds for the deci-

sion-making stage.

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In studies on MOGs, selecting a strategy is based on one out of three different representa-

tion types of the strategy's set of performance vectors. Namely, either by a scalar, a unique

vector in the objective space, or by a sub-set in the same objective space.

In most studies on MOGs, such as in [1]–[4], [7]–[15], solving the problem involves the

aggregation of payoffs by a weighted sum approach. In such studies, strategy representation

belongs to the first type. Studies, using the second and third types of strategy representation,

are rare. In works [3], [16], [17] strategy representation belongs to the second type. In these

studies the unique representation vector is the nadir point.

To the best of our knowledge, only [18] and [19] employ the third type of strategy repre-

sentation.

These two works employ a set-based worst-case optimization approach. The main differ-

ence between [19] and [18] is that in [18] the game is solved by finding the rationalizable

strategies from the perspective of one player only. In contrast, while at [19] the MOG with

postponed objective preferences is considered from the perspectives of both players. In addi-

tion, [19] provides an answer to the non-trivial question on how a rational player, which has

no knowledge of the opponent's preferences toward objectives and has made the self-decision

to postpone preferences of objectives, should be defined.

The set of strategies may be finite or infinite. When the sets of strategies are finite then the

set of all possible payoffs is finite too. In such case, each player has to examine a finite set of

strategies based on a finite set of payoffs. The procedure of finding the rationalizable strate-

gies is the main subject of ours previous work [19]. In the case of finite set of strategies, this

searching procedure supposed to yield a unique subset of rationalizable strategies for each

player.

However, this is not the situation in the case when the set of strategies is infinite or even if

the set is of large cardilality. In such instance it is impossible to compute all the payoffs of all

the potential interactions and therefore a different solution technique is required.

The papers [17], [18] and [20] deal with MOGs involving sets of infinite strategies. The

strategies are continuous variables; hence there are an infinite possible interactions and pay-

offs. The solutions are obtained by means of evolutionary algorithms. In papers [17], [18], an

genetic algorithm (NSGAII) is used in order to find the rationalizable strategies of each player

at a time. In [20], the problem was solved by co-evolutionary algorithm and the rationalizable

strategies of both players were find simultaneously. In both case, the search for rationalizable

strategies is of a stochastic nature.

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In the case of set of infinite (continence) strategies, it is unlikely that two different MOG's

solution will yield the exact same rationalizable strategies. Therefore it is impossible to com-

pare between two solutions obtained by different algorithms or even different runs of the

same algorithm. Moreover, the fitness of individuals in each solution is the results obtained

out of interaction with a different opponent population, meaning that not only that the evalu-

ated strategies are different, their payoff is obtained by interacting with different opponents.

The two facts described above generate the need to a measure that will allow such a compari-

son.

2 Test functions for MOGs

One of the ways to evaluate an algorithm is to use test function (benchmark). These func-

tions are known as artificial landscapes and are useful in the evaluation of optimization algo-

rithms characteristics, such as velocity of convergence or solution precision. The test func-

tions are chosen in order to demonstrate different situations that optimization algorithms have

to cope with, such as non-convex functions, discontinuity etc.

For MOO there exist in literature test functions (see [21]–[23]). These functions are useful

to evaluate characteristics of optimization algorithms. However, for MOGs as formulated

here, we could not find any test case in the surveyed papers. Therefore there is need to formu-

late such functions that will be used later in the evaluations of the algorithms performances.

The next table in section .שגיאה! מקור ההפניה לא נמצא presents 20 objectives function. Any

combination of two functions form a different MOG as presented in the following table at

section .שגיאה! מקור ההפניה לא נמצא.

These MOGs are destined to exhibit different types of sets of rationalizable strategies. In

these examples, the ranges of possible strategies are limited. Therefore it is possible to map

each MOG payoffs by choosing a finite and discreet set of strategies evenly distributed in-

stead the continuance (and infinite) range. This will allow us to solve the problem by a single

iteration and to obtain the sets of rationalizable strategies for each player in each MOG. These

sets of rationalizable strategies will serve as the reference "exact" solution in order to evaluate

the solutions obtained by different algorithms.

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3 Notations

3.1 Basic definition in MOGs

In this work a game between two players, 𝑃1 and 𝑃2, is considered. Each player may

choose a strategy out of a set of strategies. Each strategy is a decision vector such that the

𝑖𝑡ℎ strategy of 𝑃1 is: 𝑠1𝑖 = [𝑠1

𝑖(1), 𝑠1

𝑖(2), … , 𝑠1

𝑖(𝑛1), … , 𝑠1

𝑖(𝑁1)] and the 𝑗𝑡ℎ strategy of 𝑃2 is:

𝑠2𝑗

= [𝑠2𝑗(1)

, 𝑠2𝑗(2)

, … , 𝑠2𝑗(𝑛2)

, … , 𝑠2𝑗(𝑁2)

] where 𝑁1, and 𝑁2 are the corresponding number of

strategy's decision parameters.

Let 𝑆1 and 𝑆2 be the sets of all possible strategies of the two players respectively,

such that: 𝑆1 = {𝑠11, 𝑠1

2, … , 𝑠1𝑖 , … , 𝑠1

𝐼} and 𝑆2 = {𝑠21, 𝑠2

2, … , 𝑠2𝑗, … , 𝑠2

𝐽} where 𝑆1 ∈ Γ ⊆

ℝ𝐼×𝑁1 and 𝑆2 ∈ Ω ⊆ ℝ𝐽×𝑁2 , where I and J are the total number of strategies of P1 and

P2, respectively. The interaction between the 𝑖𝑡ℎ strategy and the 𝑗𝑡ℎ strategy played by

𝑃1and 𝑃2, respectively, results in the following game:

𝑔𝑖,𝑗 ∶= {𝑠1𝑖 , 𝑠2

𝑗} ∈ Φ ⊆ ℝ𝑁1 × ℝ𝑁2 (1)

All of the alternative interactions, among all strategies of both players, form the set G of

all possible games:

𝐺 = {𝑔𝑖,𝑗 ∈ 𝛷 | 𝑠1𝑖 ∈ 𝑆1 ∧ 𝑠2

𝑗∈ 𝑆2} (2)

Assuming a zero-sum game, each game 𝑔𝑖,𝑗 is evaluated using an objective vector of

performances (payoff vector), 𝑓̅ = [𝑓(1), 𝑓(2), … , 𝑓(𝑘), … , 𝑓(𝐾)]𝑇

∈ Ψ ⊆ ℝ𝐾. More specifi-

cally, the result of the 𝑔𝑖,𝑗 game is assessed by:

𝑓�̅�,𝑗 = [𝑓𝑖,𝑗(1)

, 𝑓𝑖,𝑗(2)

, … , 𝑓𝑖,𝑗(𝑘)

, … , 𝑓𝑖,𝑗(𝐾)

]𝑇

∈ 𝛹 ⊆ ℝ𝐾 (3)

The set of all the interaction between a strategy 𝑠1𝑖 of player 𝑃1 and all of the availa-

ble strategies of the second player 𝑃2 (𝑠2𝑗 ∀ 𝑗 = 1, … , 𝐽) is:

𝑔𝑠1𝑖 = {𝑔𝑖,1, … , 𝑔𝑖,𝑗 … , 𝑔𝑖,𝐽} (4)

In the same way, the set of all the interaction between a strategy 𝑠2𝑗 of player 𝑃2 and

all of the available strategies of the second player 𝑃1 (𝑠1𝑖 ∀ 𝑖 = 1, … , 𝐼 ) is: 𝑔

𝑠2𝑗 =

{𝑔1,𝑗, … , 𝑔𝑖,𝑗 … , 𝑔𝐼,𝑗}.

The set of payoff vectors that represent the performances of strategy 𝑠1𝑖 is the set:

𝐹𝑠1𝑖 = {𝑓�̅�,1, … , 𝑓�̅�,𝑗, … , 𝑓�̅�,𝐽} (5)

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In the same way, the set that represent the performances of strategy 𝑠2𝑗 of the maximizer

is the set:

𝐹𝑠2

𝑗 = {𝑓1̅,𝑗, … , 𝑓�̅�,𝑗, … , 𝑓�̅�,𝑗} (6)

4 Methodology

There are many methods to compare between two solutions obtained by different algo-

rithms or different runs of the same algorithm. These methods can be divided into two catego-

ries.

The first comparison category is of the methods being conducted in the strategies' space

(design space) and the second is being conducted in the objective space.

In both types of comparison, the results of the algorithms will be compared, using the "ex-

act" solution as described in section ‎2. This solution will be referred as the "reference solu-

tion"

4.1.1 Notations of the reference solution

Let the finite set of strategies of 𝑃1 and 𝑃2 in the reference solution be 𝑆1 and 𝑆2 respec-

tively. The number of all possible strategies, in the reference solution, of players 𝑃1 is

𝐼 = |𝑆1| and of players 𝑃2 is 𝐽 = |𝑆2| .

Let the set of rationalizable strategies of 𝑃1 in reference solution be 𝑆1∗ ⊆ 𝑆1 and the set of

rationalizable strategies strategies of 𝑃2 in the reference solution be 𝑆2∗ ⊆ 𝑆2. The number of

rationalizable strategies of players 𝑃1 and 𝑃2 in the reference solution are the cardinality of the

sets 𝐼∗ = |𝑆1∗| ⊆ 𝐼 and 𝐽∗ = |𝑆2

∗| ⊆ 𝐽 respectively and the indexes of the rationalizable strate-

gy in these sets are 𝑖∗ ∈ 𝐼∗ ⊆ 𝐼 and 𝑗∗ ∈ 𝐽∗ ⊆ 𝐽 respectively. The members in the set of ra-

tionalizable strategies of 𝑃1 and 𝑃2 are 𝑠1𝑖∗

∈ 𝑆1∗ and 𝑠2

𝑗∗

∈ 𝑆2∗ respectively.

4.1.2 Notations of the algorithms' solution

All the proposed algorithms are Evolutionary Algorithms (EAs). The design parameters

are the strategies and the number of strategies, at each iteration, is the population's size. Un-

like the reference solution, here, although the cardinality of the set of strategies (the size of

the population) at each iteration is finite, the set of all possible strategies is infinite.

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Let the set of strategies of 𝑃1 and 𝑃2 in the first algorithm solution be 𝑆1(1)

and 𝑆2(1)

re-

spectively. The cardinalities of these two sets are 𝐼(1) = |𝑆1(1)

| and 𝐽(1) = |𝑆2(1)

|. Note that 𝐼(1)

and 𝐽(1) are equal to the population size of players 𝑃1 and 𝑃2 in the first algorithm solution. In

the same way, 𝑆1(2)

and 𝑆2(2)

are the set of strategies of 𝑃1 and 𝑃2 in the second algorithm solu-

tion and their cardinalities are 𝐼(2) = |𝑆1(2)

| and 𝐽(2) = |𝑆2(2)

|.

Let the set of rationalizable strategies of 𝑃1 in the first algorithm solution be 𝑆1∗(1)

⊆ 𝑆1(1)

and the set of rationalizable strategies of 𝑃2 in the solution be 𝑆2∗(1)

⊆ 𝑆2(1)

. The numbers of

rationalizable strategies of players 𝑃1 and 𝑃2 in this solution are: 𝐼∗(1) = |𝑆1∗(1)

| ⊆ 𝐼(1) and

𝐽∗(1) = |𝑆2∗(1)

| ⊆ 𝐽(1) respectively. In the same way, the set of rationalizable strategies of 𝑃1

and 𝑃2 in the second algorithm solution are 𝑆1∗(2)

⊆ 𝑆1(2)

and 𝑆2∗(2)

⊆ 𝑆2(2)

, and the number of

rationalizable strategies are: 𝐼∗(2) = |𝑆1∗(2)

| ⊆ 𝐼(2) and 𝐽∗(2) = |𝑆2∗(2)

| ⊆ 𝐽(2) respectively.

The indexes of the rationalizable strategies in the first solution are 𝑖∗(1) ∈ 𝐼∗(1) ⊆ 𝐼(1) and

𝑗∗(1) ∈ 𝐽∗(1) ⊆ 𝐽(1) and of the second solution are 𝑖∗(2) ∈ 𝐼∗(2) ⊆ 𝐼(2) and 𝑗∗(2) ∈ 𝐽∗(2) ⊆ 𝐽(2).

The members in the set of rationalizable strategies of 𝑃1 and 𝑃2 of the first algorithm solution

are 𝑠1𝑖∗(1)

∈ 𝑆1∗(1)

and 𝑠2𝑗∗(1)

∈ 𝑆2∗(1)

and of the second solution 𝑠1𝑖∗(2)

∈ 𝑆1∗(2)

and 𝑠2𝑗∗(2)

∈ 𝑆2∗(2)

respectively.

4.1.3 Notations of strategy performances in mixture of reference and solu-

tion strategies

The equations (5 and (6 define the set of payoff vector of a single strategy of the minimiz-

er and maximizer respectively. However, these sets describe the performances set within the

same solution. Namely, strategy 𝑠1𝑖 is evaluated by all its interactions with the maximizer's

strategies within the same solution. Though, here there is a need to define the strategy perfor-

mances when interacting with a set of maximizer's strategies from a different solution.

The set of all the available strategies of the minimizer in solution 𝑞 is the set: 𝑆1(𝑞)

=

{𝑠11(𝑞)

, 𝑠12(𝑞)

, … , 𝑠1𝑖(𝑞)

, … , 𝑠1𝐼(𝑞)

} and the set of all the available strategies of the maximizer in

solution 𝑝 is: 𝑆2(𝑝)

= {𝑠21(𝑝)

, 𝑠22(𝑝)

, … , 𝑠2𝑗(𝑝)

, … , 𝑠2𝐽(𝑝)

}.

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The interaction between the 𝑖𝑡ℎ strategy of the minimizer in the solution 𝑞, 𝑠1𝑖(𝑞)

and the

𝑗𝑡ℎ strategy the maximizer in the solution 𝑝, 𝑠2𝑗(𝑝)

, results in the following game:

𝑔 (𝑠1𝑖(𝑞)

, 𝑠2𝑗(𝑝)

). Note that in this representation the notation 𝑔𝑖,𝑗 is rewritten as 𝑔(𝑠1𝑖 , 𝑠2

𝑗).

The set of all the interaction between a strategy 𝑠1𝑖(𝑞)

of the minimizer in the solution 𝑞

𝑞 and all of the available strategies of the second the maximizer in a different solution 𝑝

𝑆2(𝑝)

is:

𝑔 (𝑠1𝑖(𝑞)

, 𝑆2(𝑝)

) = {𝑔 (𝑠1𝑖(𝑞)

, 𝑠21(𝑝)

) , … , 𝑔 (𝑠1𝑖(𝑞)

, 𝑠2𝑗(𝑝)

) , … , 𝑔 (𝑠1𝑖(𝑞)

, 𝑠2𝐽(𝑝)

)} (7)

In the same way, the set of all the interaction between a strategy 𝑠2𝑗(𝑞)

of the maximizer

in the solution 𝑞 and all of the available strategies of the second the minimizer in a different

solution 𝑝 𝑆1(𝑝)

is:

𝑔 (𝑆1(𝑝)

, 𝑠2𝑗(𝑞)

) = {𝑔 (𝑠11(𝑝)

, 𝑠2𝑗(𝑞)

) , … , 𝑔 (𝑠1𝑖(𝑝)

, 𝑠2𝑗(𝑞)

) , … , 𝑔 (𝑠1𝐼(𝑝)

, 𝑠2𝑗(𝑞)

)} (8)

The result of the 𝑔 (𝑠1𝑖(𝑞)

, 𝑠2𝑗(𝑝)

) game is assessed by the payoff vector 𝑓̅ (𝑠1𝑖(𝑞)

, 𝑠2𝑗(𝑝)

).

The set of payoff vectors that represent the performances of strategy 𝑠1𝑖(𝑞)

when interact-

ing with 𝑆2(𝑝)

is the set:

𝐹 (𝑠1𝑖(𝑞)

, 𝑆2(𝑝)

) = {𝑓̅ (𝑠1𝑖(𝑞)

, 𝑠21(𝑝)

) , … , 𝑓̅ (𝑠1𝑖(𝑞)

, 𝑠2𝑗(𝑝)

) , … , 𝑓̅ (𝑠1𝑖(𝑞)

, 𝑠2𝐽(𝑝)

)} (9)

The set of payoff vectors that represent the performances of strategy 𝑠2𝑗(𝑞)

when interact-

ing with 𝑆1(𝑝)

is the set:

𝐹 (𝑆1(𝑝)

, 𝑠2𝑗(𝑞)

) = {𝑓̅ (𝑠11(𝑝)

, 𝑠2𝑗(𝑞)

) , … , 𝑓̅ (𝑠1𝑖(𝑝)

, 𝑠2𝑗(𝑞)

) , … , 𝑓̅ (𝑠1𝐼(𝑝)

, 𝑠2𝑗(𝑞)

)} (10)

4.2 Comparing in the design space

The comparison between two algorithms' solutions in the design space is based on Euclid-

ian distance between each algorithm solution and the reference solution. For example, the Eu-

clidian distance between a strategy of the reference and a strategy of the first algorithm solu-

tion of player 𝑃1 is 𝑑 (𝑠1𝑖∗

, 𝑠1𝑖∗(1)

). The distance of solution 𝑆1∗(1)

from reference strategy 𝑠1𝑖∗

is

the minimal distance:

𝑑(𝑠1𝑖∗

, 𝑆1∗(1)

) = min𝑖∗(1)∈𝐼∗(1) {𝑑 (𝑠1𝑖∗

, 𝑠1𝑖∗(1)

)} (11)

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The distance of solution 𝑆1∗(1)

from reference solution 𝑆1∗is the minimal distance:

𝑑(𝑆1∗, 𝑆1

∗(1)) = max𝑖∗∈𝐼∗ {𝑑(𝑠1

𝑖∗, 𝑆1

∗(1))} (12)

In the same way the distance of solution 𝑆2∗(1)

from reference strategy 𝑠2𝑗∗

is the minimal dis-

tance:

𝑑 (𝑠2𝑗∗

, 𝑆2∗(1)

) = min𝑗∗(1)∈𝐽∗(1) {𝑑 (𝑠2𝑗∗

, 𝑠2𝑗∗(1)

)} (13)

The distance of solution 𝑆2∗(1)

from reference solution 𝑆2∗is the minimal distance:

𝑑(𝑆2∗, 𝑆2

∗(1)) = max𝑗∗∈𝐽∗ {𝑑 (𝑠2

𝑗∗

, 𝑆2∗(1)

)} (14)

The distance between first algorithm and the reference solution is:

𝑑(1) = max𝑞=1,2 {𝑑(𝑆𝑞∗, 𝑆𝑞

∗(1))} (15)

The reference solution is assumed to be the exact solution; hence the algorithm with the

smallest distance from the reference solution (index q) is the better one:

𝑑(𝑞) < 𝑑(𝑝) ⟹ 𝑑(𝑞) ≻ 𝑑(𝑝) 𝑓𝑜𝑟 𝑝, 𝑞 = 1,2 𝑎𝑛𝑑 𝑝 ≠ 𝑞 (16)

4.3 Comparing in the objective space

The two solutions obtained by different runs or algorithms are four sets of rationalizable strat-

egies one for each player at each solution. Since the GA is a stochastic search one cannot ex-

pect that the sets will contain the same number of strategies. Moreover, in the objective space

the distance measure is no longer valid. Since the performances of a strategy is not a single

vector but a set of vectors it raise the question how to camper the two solutions? Which one of

them is closer to the reference solution? In other word, the question is how to evaluate the

similarity of two sets of vectors.

Reference

solution Algorithmic solution

Algorithm 1 Algorithm 2

𝑆1∗(2)

𝑆2∗(2)

𝑆1∗(1)

𝑆2∗(1)

𝑆1∗ 𝑆2

∗

𝑠1𝑖∗

𝑠2𝑗∗

𝑠1𝑖∗(1)

𝑠2𝑗∗(1)

𝑠1𝑖∗(2)

𝑠2𝑗∗(2)

Page 9 of 12

The comparison:

Conceder a reference solution that includes the minimizer set of rationalizable strategies 𝑆1∗

and 𝑆2∗ for the minimizer and the maximizer respectively. These sets are depicted by black

(maximizer) and gray (minimizer) stares in Figure 1. The cardinalities of these set are:

𝐼∗ = |𝑆1∗| = 8 and 𝐽∗ = |𝑆2

∗| = 6. Also, conceder, two algorithms' solutions denoted by circles

and squares. The cardinalities of the sets of rationalizable strategies of the minimizer and

maximizer in both solutions are: 𝐼1∗ = |𝑆1

∗(1)| = 6, 𝐽1

∗ = |𝑆2∗(1)

| = 5, 𝐼2∗ = |𝑆1

∗(2)| = 4, 𝐽2

∗ =

|𝑆2∗(2)

| = 5 as represented by gray and black circles and gray and black squares respectively

in Figure 1.

𝑠1𝑖∗

𝑆2∗(2)

𝑆2∗(1)

𝑠1𝑖∗

𝑆1∗(2)

𝑆1∗(1)

𝑠2𝑗∗

𝑠2𝑗∗

Figure 2: One iteration of comparison. One comparison of the maximizer and the

minimizer solutions on the left and right panels respectively.

Page 10 of 12

The evaluation of the two solutions is done by comparing their rationalizable strategies (for

the minimizer and the maximizer) when interacting with a strategy of the reference set. More

specifically, the next interactions are performed and evaluated:

First the score of the first and second solution is set on zero: 𝑠𝑐𝑜𝑟𝑒1 = 𝑠𝑐𝑜𝑟𝑒2 = 0. Next the

set of payoff vector 𝐹(𝑠1𝑖∗, 𝑆2

∗(1)) is compared to the set 𝐹(𝑠1

𝑖∗, 𝑆2∗(2)

). This comparison is re-

peated for all the minimizer strategies in the reference solution. Each time, the anti-optimal

fronts (Pareto front of the maximization problem) of the two sets are compared. These anti-

optimal fronts are denoted as: 𝐹−∗(𝑠1𝑖∗, 𝑆2

∗(1)) and 𝐹−∗(𝑠1

𝑖∗, 𝑆2∗(2)

).

If 𝐹−∗(𝑠1𝑖∗, 𝑆2

∗(1)) ≻𝑚𝑎𝑥 𝐹−∗(𝑠1

𝑖∗, 𝑆2∗(2)

) → 𝑠𝑐𝑜𝑟𝑒1 = 𝑠𝑐𝑜𝑟𝑒1 + 1

If 𝐹−∗(𝑠1𝑖∗, 𝑆2

∗(2)) ≻𝑚𝑎𝑥 𝐹−∗(𝑠1

𝑖∗, 𝑆2∗(1)

) → 𝑠𝑐𝑜𝑟𝑒2 = 𝑠𝑐𝑜𝑟𝑒2 + 1

If 𝐹−∗(𝑠1𝑖∗, 𝑆2

∗(2)) ∼ 𝐹−∗(𝑠1

𝑖∗, 𝑆2∗(1)

) (there are non-dominated sets) then none of them

gets a point.

After comparing the performances of the maximizers' strategies, this procedure is carried

out for the minimizers rationalizable strategies. Specifically, each strategy of the maxi-

mizer in the reference solution, 𝑠2𝑗∗

is confronted with the sets of the rationalizable strate-

gies of the minimizers in the two solutions, 𝑆1∗(1)

and 𝑆1∗(2)

. Again

If 𝐹−∗(𝑆1∗(1)

, 𝑠2𝑗∗

) ≻𝑚𝑖𝑛 𝐹−∗(𝑆1∗(2)

, 𝑠2𝑗∗

) → 𝑠𝑐𝑜𝑟𝑒1 = 𝑠𝑐𝑜𝑟𝑒1 + 1

If 𝐹−∗(𝑆1∗(2)

, 𝑠2𝑗∗

) ≻𝑚𝑖𝑛 𝐹−∗(𝑆1∗(1)

, 𝑠2𝑗∗

) → 𝑠𝑐𝑜𝑟𝑒2 = 𝑠𝑐𝑜𝑟𝑒2 + 1

If 𝐹−∗(𝑆1∗(1)

, 𝑠2𝑗∗

) ∼ 𝐹−∗(𝑆1∗(2)

, 𝑠2𝑗∗

) (there are non-dominated sets) then none of

them gets a point.

After all pairwise comparisons have been conducted, each solution has a score, the one

with the highest score is the better one.

Page 11 of 12

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