1.1 Motivating challenges

Understanding the emergence of strategies within complex collective systems poses significant challenges for engineers and system designers when evaluating a system’s ability to integrate with others and align with common goals. This property, referred to as directional integrability [14], when extended to larger systems-of-systems, is linked to each subsystem’s level of managerial control and collaboration [7]. Individual actors within the system face critical choices regarding the openness of their subsystems, and these choices have far-reaching implications for the system’s overall performance. The decision to pursue an open strategy, facilitating full integration, or to maintain a closed system introduces complex trade-offs. Actors must weigh the potential benefits of collective integration against the costs and risks associated with opening their own systems. Although an open strategy might be optimal on a collective level, it might not be the most advantageous path for the individual actor.

The complexity escalates dramatically as the number of actors within the system increases. Decision-making becomes a tangled web of anticipating other actors’ choices, potential coalitions, and the risks associated with incomplete information. Understanding the social dynamics at play becomes essential, as factors like communication, the way options are framed [15, 16], and actors’ varying motivations and incentives [17] significantly influence the strategic landscape. A particularly worrying outcome in multi-actor systems is the potential for integration to halt. Imagine a scenario where a single actor initially takes the risk of opening their system, hoping others will follow. However, if others exploit this openness, keeping their systems closed and reaping the benefits without contributing, this can discourage further integration efforts. The system can easily become stuck in a situation where the benefits of wider integration seem out of reach, as no individual actor wants to be the only one taking on the burden of an open system.

To overcome these challenges, there’s a critical need for new modeling approaches and analytical tools explicitly designed to handle the emergent dynamics of multi-actor scenarios. These models must move beyond traditional two-player game-theoretic frameworks to robustly capture complex interactions and coalition formation. Holistic decision-making approaches considering the technical, social, and strategic interdependencies are needed. Alongside sophisticated modeling, a focus on developing mechanisms to incentivize integration is paramount. Finding ways to align individual actor benefits with the goals of the collective system is essential to ensure sustained progress and avoid stagnation. Addressing these challenges is crucial for successfully engineering and managing complex systems-of-systems. A deep understanding of these emergent strategic dynamics and incentive structures will empower us to build systems that foster coordination, reduce incompatibilities, and ultimately facilitate the harmonious integration of interdependent subsystems for the greater good.

1.2 Our research endeavor

Our work focuses on studying complex strategy dynamics in collective systems, with an aim to comprehend the factors that influence the decisions and interactions of system actors. We acknowledge the complexity of these systems and strive to design mechanisms that promote collaboration in large-scale systems-of-systems. The objective of this paper is to provide a general mathematical characterization and visual representation of emergent strategy dynamics in binary games that can be leveraged in the strategic design of engineering systems. It extends the existing concept of structural fear and greed in two-player games to any finite number of actors. These structural factors are presented on a Cartesian coordinate plane, with each quadrant defining a dynamical domain with specific strategic stability conditions.

The remainder of this paper is organized as follows. Section 2 recaps the mainstream treatment of strategy dynamics centered around normal-form games with two players and two available strategies each. Section 3 presents our extended characterization of strategy dynamics for binary strategic-form games with any number of players, with more details provided in the S1 Appendix. Here we introduce the concept of strategic hindrance, with Section 3.3 applying the proposed framework to assess the evolution of strategy dynamics in the design for integrability of an urban transit system stylized as a volunteer’s dilemma, and Section A.4 in S1 Appendix extending this framework to visually assess additional generic strategic settings (viz. public goods game, fated truel game, majority game, and others). Then, in Section 4, we address the assumptions and limitations of our proposed approach. We conclude with Section 5, summarizing contributions and our vision for future work.

2.1 Normal-form games

A normal-form game consists of

• A player set consisting of n ≥ 1 player names or other identifiers (ids);
  • for the sake of simplicity, we use and dummy ids ;
  • a game with n = 1 is referred to as a “trivial” decision problem.
• A finite set of pure individual strategies , with ;
  • a pure strategy is a complete contingency plan that a player can execute in anticipation of the actions that the other players could take;
  • the product is the set of all possible pure collective strategies s = 〈s₁, …, s_n〉, with ;
  • an n × 2 normal-form game (also known as binary game) is one in which m_i = 2 for every , resulting in a total of 2ⁿ pure collective strategies.
• A payoff function valuing the preference of player i for each ;
  • the value of U_i(s) is assumed to be measured on a cardinal utility scale;
  • , or equivalently, max_s|U_i(s)| ≠ 0 for every .

The central objective of game theory is to understand what will, could, or should happen in a given game. One of the main solution concepts in game theory is the Nash equilibrium, which unified several earlier ideas related to the stability of collective action in non-cooperative games [18]. The ideas we develop in this work are relevant to studying pure-strategy Nash equilibria (PNE), in which every player is expected to commit fully to one and only one pure individual strategy (see Definition A.1 in the S1 Appendix).

Fig 1(a) and 1(b) show examples of individual payoff matrices in 2 × 2 social dilemmas. We can also write all payoffs in a 2 × 2 game as a simplified bimatrix: (1) where for every player . The collective strategy set is . Strategies φ = 〈φ_i, φ_j〉 and ψ = 〈ψ_i, ψ_j〉 are called diagonal; the other collective strategies, namely those in set , are called off-diagonal. Labeling a combination of individual strategies as diagonal or off-diagonal is left to the reader’s discretion, and it does not affect the nature of the game. In this work, φ is also referred to as the status quo strategy—or starting point—and ψ is treated as the alternative or desired outcome (from the perspective of the incentive designer or researcher) consistent with economics literature [19].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Generic individual payoff matrix in a two-player two-strategy social dilemma game in normal form.

The strategy labels φ and ψ represent “defection” and “cooperation”, respectively. The value of P represents the punishment resulting from mutual defection; R > P is the reward of mutual cooperation; S stands for sucker’s payoff (what a player gets from cooperating if the other defects); and T stands for temptation to defect (the incentive to deviate away from the cooperative strategy in the expectation of higher returns).

https://doi.org/10.1371/journal.pone.0301394.g001

2.2 Structural fear and greed in 2 × 2 social dilemmas

Classical 2 × 2 social dilemmas name each strategy after the description that human players would use to describe their behavior [15] and are often symmetric—meaning their payoff functions do not depend on the players’ identities, i.e. . For instance, let us assume that φ_k is a “defective” individual strategy and that ψ_k is a “cooperative” one. Coordinating on φ and ψ can thus be called mutual (or unanimous) defection and cooperation, respectively. These are the usual strategy labels in a classical social dilemma. Fig 1(a) shows a generic payoff matrix used to characterize these games. Cooperation yields a “reward”, U_i(ψ_i, ψ_j) = R, which would be preferable to mutual defection, which results in a “punishment”, U_i(φ_i, φ_j) = P < R [20]. The values U_i(ψ_i, φ_j) = S and U_i(φ_i, ψ_j) = T are called “sucker’s” and “temptation” payoffs, respectively, terms borrowed from the prisoner’s dilemma (PD) game [21]. A sample individual payoff matrix of a PD game is shown in Fig 1(b). A PD game is characterized by the relationships T > R > P > S. The stag hunt game, in contrast, of which a sample individual payoff matrix is presented in Fig 1(c), satisfies R > T > P > S [22]—a “periodic table” of 2 × 2 games can be derived from different orderings of P, R, S, and T as described in Ref. [23].

The S payoff is associated with a player’s “fear” of pursuing cooperation when others might defect. Meanwhile, the T payoff is a player’s incentive to defect away from mutual cooperation driven by the “greed” for higher returns. Ahn et al. [24] measure structural fear and greed, respectively, as (2) and (3) The expressions “P − S” and “T − R” in Eqs (2) and (3) are the deviation losses measured with respect to the status quo strategy φ_i. In the context of the 2 × 2 PD game, the structural fear (F_i > 0) is the deficit of unilaterally cooperating when the other defects and the structural greed (G_i > 0) is the benefit of taking advantage of a cooperating partner by choosing to defect instead [25]. Either factor favors defection if positive. The higher the values of F_i and G_i, the greater the incentive that player i has to defect instead of playing a cooperative strategy.

Previous works have identified four 2 × 2 strategy dynamical domains exhibiting different payoff and risk dominance conditions based on the values of S and T, while setting P = 0 and R = 1 [26–29]. These values map to one combination of positive (high) or negative (low) structural fear and greed, as provided below and in Fig 1, per Eqs (2) and (3):

Harmony / HA (or cooperation):

Coexistence / CX (or anti-coordination):

Bistability / BI (or coordination):

Defection / DE: We introduce a fifth dynamical domain to represent player i’s indifference about the strategy set: Indifference / ZZ: The 2 × 2 harmony and defection dynamical domains are also known as (type ψ or type φ) dominance dynamics [30, 31]; while the coexistence and bistability domains are sometimes referred to as negative and positive frequency dependence dynamics, respectively [32, 33]. The latter are also known as bipolar [34]—note that any 2 × 2 coexistence game can be transformed into a bistability/bipolar game by switching the strategy labels of one of the two players.

Fig 2 shows how the values of S and T map to the factors F_i and G_i per Eqs (2) and (3), assuming P = 0 and R = 1. The vast majority of the literature on social dilemma games focuses on the payoff region defined by S ∈ [−1, 1] and T ∈ [0, 2]. These games cover the entire harmony domain, ⅜ of the bistability and coexistence domains, and only ⅙ of the defection domain—that is, of the whole fear and greed strategy dynamics space. The structural fear and greed values for the PD game in Fig 1(a) are F_i = G_i = 1/3, which fall on defection dynamics, while the values for the SH game in Fig 1(b) are F_i = 1/3 and G_i = −1/3, falling on bistability dynamics. Examples of harmony and coexistence games respectively include the concord game (S = 1/3 and T = 2/3 ⇒ F_i = G_i = −1/3) and the chicken game (S = 1/2 and T = 3/2 ⇒ F_i = −1/3 and G_i = 1/3).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Mapping between the sucker’s (S) and the temptation (T) payoffs and the structural fear (F_i) and structural greed (G_i) factors per Eqs (2) and (3), assuming P = 0 and R = 1.

Each quadrant is labeled after the 2 × 2 strategy dynamics resulting from different combinations of low and high levels of structural fear and greed: HA / harmony (F_i < 0, G_i < 0); CX / coexistence (F_i < 0, G_i > 0); BI / bistability (F_i > 0, G_i < 0); and DE / defection (F_i < 0, G_i > 0). The vast majority of the literature on social dilemma games focuses on the shaded region defined by S ∈ [−1, 1] and T ∈ [0, 2].

https://doi.org/10.1371/journal.pone.0301394.g002

2.3 Deviation losses and strategic stability

For general n × 2 games, let us define the rescaled deviation loss ℓ_i in terms of the difference in player i’s payoff when deviating from to assuming the strategies of the other players remain fixed:

Definition 1 Let be a standpoint player in the n-player normal-form game where is the set of n players, is the set of individual strategies, are the collective strategies, and is player i’s payoff function. The rescaled individual payoff loss of deviating from to contingent on a collective strategy s_−i played by all other players () is (4) where A_i = max_s U_i(s) − min_s U_i(s) > 0 is the peak-to-peak amplitude of i’s payoffs.

If is an n × 2 game and , we have that ℓ_i(φ_i, s_−i) = −ℓ_i(ψ_i, s_−i); for the sake of simplicity, we write moving forward. Thus, the maximum number of unique values of per player in a binary normal-form game is 2ⁿ⁻¹. For n = 2, we write (5) We now rewrite Eqs (2) and (3) in terms of Eqs (4) and (5) as (6) and (7) If the labels φ and ψ are swapped above, F_i(ψ, φ) = −G_i(φ, ψ) and G_i(ψ, φ) = −F_i(φ, ψ).

In binary games, equilibrium conditions depend only on the signa of each player’s loss of deviating away from a collective strategy (see Definitions A.2 and A.2a in the S1 Appendix). More precisely, is a PNE if In a 2 × 2 game, if both players’ structural fear or greed values fall in the same region (as is the case for symmetric games), then (8) for some and some , where is the signum function. If both sides of Eq (8) are equal to +1, then s = 〈s_i, s_j〉 must be a strict PNE. We denote the set of PNE as . We use the bimatrix notation in Eq (1) to compare cardinal payoffs with the signa of the rescaled deviation losses and evaluate in a 2 × 2 game: (9) Mapping a 2 × 2 game to the bimatrix form in Eq (9) allows us to identify any PNE in it more quickly and how they relate to the structural fear and greed factors. Fig 3 reintroduces the main 2 × 2 strategy dynamical domains using the aforementioned notation. As an example, the payoffs in a symmetric game with harmony dynamics can be written as we underline the cells where each of the two signa are greater than or equal to zero. Thus in the game above, there is one PNE, ψ, since F_i, F_j < 0 and G_i, G_j < 0, and implies that both players would always prefer ψ_i (second row) and ψ_j (second column) over φ_i and φ_j. Similarly, in a symmetric game with coexistence dynamics, for example there are two PNE, = {〈φ_i, ψ_j〉, 〈ψ_i, φ_j〉}, the off-diagonal strategies, since F_i < 0 and G_j > 0—likewise, F_j < 0 and G_i > 0—imply indicating that not taking the same action would be the most economically rational outcome.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Classification of strategy dynamics in 2 × 2 social dilemma games in terms of the individual payoffs U_i(s_i, s_j) and their relative differences (+ versus −).

An economically rational player would presumably prefer those collective strategies marked with a plus sign (+) over those marked with a minus sign (−). In the case of indifference dynamics, notice that both s_i rows are equal, implying that player i’s payoff only depends on player j’s selection.

https://doi.org/10.1371/journal.pone.0301394.g003

A game with one or more PNE can also be formed by two individual payoff matrices exhibiting different strategy dynamics. We provide two examples. One: we can model U_i after the harmony payoffs in Fig 3(a) and U_j after the (transposed) defection payoffs in Fig 3(d); using the bimatrix game notation in Eq (9), we can write (10) where the collective strategy 〈ψ_i, φ_j〉 is the only equilibrium in this game. The individual strategies in Eq (10) are also perfectly limited, meaning they yield an individual payoff that depends only on each player’s strategy [35]. Using Eqs (6) and (7), we calculate the structural fear and greed for players i and j as F_i = G_i = −1 < 0 (both factors support ψ_i) and F_j = G_j = +1 > 0 (both factors support φ_j), respectively. Two: let us reuse the coexistence payoffs to model U_i and (transposed) bistability payoffs per Fig 3(c) to model U_j and write (11) this time, however, there are no PNE. Also, each player only has one perfectly limited strategy: φ_i and φ_j always yield a zero payoff of zero to players i and j regardless of the other’s choice of individual strategy. The values of structural fear and greed for players i and j are 〈F_i, G_i〉 = 〈−1/2, +1/2〉 and 〈F_j, G_j〉 = 〈+1/2, −1/2〉, respectively. If only the signa of each player’s structural fear and greed factors are considered, the bimatrix game in Eq (11) is qualitatively equivalent to the 2 × 2 zero-sum game (12)

In the following section, we extend the definitions of , F_i, and G_i to normal-form games with any number of players n ≥ 2 and binary strategy sets for every .

3.1 Dissecting fear and greed in n × 2 games: Strategic hindrance

We start by defining player-reduced binary normal-form games:

Definition 2 Let be a normal-form game; let be a standpoint player in game ; let be a proper subset of players in excluding i; and let be the non-empty set of players different from i that are not in . Assume, without loss of generality, that player i anticipates a collective action by the players in . From player i’s standpoint, the normal-form game reduces to (13) That is, is a player-reduced game observed by player i within .

We use Definition 2 to reformulate Eq (4) in terms of s_i ∈ {φ_i, ψ_i} and as (14) where , ; and , .

Next, we characterize player i’s fear and greed in game assuming a coordinated action by player-set and an unmovable action by player-set :

Definition 3 Let be a player-reduced game observed by player i, where is the collective strategy of the (excluded) players in . Assume that, in the face of uncertainty, player i conjectures that all players in act as one player with individual strategy ; and that the players in have adopted a presumably immovable collective strategy of which player i is “certain.” The values of structural fear and greed specific to player i concerning a unanimous course of action by the players in are (15) and (16) where A_i = max_s U_i(s) − min_s U_i(s) > 0 and .

A few of notes on Definition 3:

• Setting —which implies and —turns Eqs (15) and (16) into Eqs (6) and (7), respectively.
• We intend the assumption of a fixed to represent the scenarios in which each player are committed to playing a particular individual strategy and player i both know of their intentions and expects them to be fully realized; we call this a type I player-reduced binary game: stationary -players.
• A second type of player-reduced binary game concerns the scenario in which one or more players in are not expected to commit to ; we call this a type II player-reduced binary game: reversing -players (see Section A.2 in the S1 Appendix for more details).

We now define the strategic hindrance of standpoint player i as the space of structural fear and greed values for all possible type I player-reduced games where no more than n − 2 players are fixed.

Definition 4 Given a binary n-player normal-form game with individual strategy set , we define the space of structural fear and greed values, or strategic hindrance, of a player as (17) where g(n) is the number of possible player-reduced binary games with stationary -players (type I).

The total number of possible type I player-reduced binary games with fixed is (see Section A.2 in the S1 Appendix): (18)

There are only 2ⁿ⁻¹ possible that can be input as one of the 2 ⋅ g(n) entries of the strategic hindrance space in Eq (4). For instance, if n = 3, g(n) = 5, and each of the 2 × 5 = 10 entries in player i’s strategic hindrance space takes one of the 2³⁻¹ = 4 values of . This implies that some deviation losses appear more than once and thus impact the emergent strategy dynamics more than others. We quantify such an impact using the function (19) which counts how many entries of each player’s strategic hindrance space are equal to based solely on the number of players in that play either φ_j and ψ_j, n_φ\i and n_ψ\i, satisfying n_φ\i + n_ψ\i = n − 1. The value of γ in Eq (19) is associated to the number of type I player-reduced binary games in Eq (18) via the following alternative formula for g(n) (see Section A.3 in the S1 Appendix): (20) where is the binomial coefficient and k counts the entries in s₋₁ that are equal to either φ_j or ψ_j—note that Eqs (19) and (20) are symmetric with respect to strategy, that is γ(n_φ\i, n_ψ\i) = γ(n_ψ\i, n_φ\i). For instance, if n = 3 and , the values of γ associated with each possible s₋₁ are and, per Eq (20), g(3) = (1/2) ⋅ [γ(0, 2) ⋅ (1) + γ(1, 1) ⋅ (2) + γ(2, 0) ⋅ (1)] = (1/2) ⋅ 10 = 5.

3.2 Strategic hindrance in 3 × 2 games

This section inspects the information that can be extracted from a strategic hindrance of a player in a 3 × 2 game when represented in a Euclidean plane. Solving Eq (17) for n = 3, player i’s strategic hindrance can be expressed as: (21) Table 1 lists each of the corresponding g(3) = 5 structural fear and greed value pairs for n = 3 in terms of the rescaled deviation losses . We use rectangular coordinates to describe a generic individual strategic hindrance space in Fig 4. In Fig 6, we add strategic hindrance spaces for the remaining two players and discuss how the visualization is useful for understanding the stability of collective actions in the game. We repeat this assessment after modifying the payoffs of one of the players, namely by multiplying by −1, which results in the strategic hindrance spaces in Fig 9.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Sample strategic hindrance space in a 3 × 2 game with g(3) = 5 unique type I player-reduced games observed by standpoint player i.

The payoff structure is shown in Fig 5.

https://doi.org/10.1371/journal.pone.0301394.g004

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Strategic hindrance space for player i in a 3 × 2 game.

https://doi.org/10.1371/journal.pone.0301394.t001

Generic individual 3 × 2 strategic hindrance space.

A player’s strategic hindrance highlights the multiple modes of biases that are baked into the payoff structure. In a one-shot game, a player may rely on simple arithmetic to play the pure strategy that maximizes their expected value. This may be the case in games with more regular payoff structures that produce fewer unique fear and greed value pairs per player, usually falling entirely within the same strategy dynamical domain—e.g., the social dilemmas analyzed in Section A.4 in S1 Appendix. All structural fear and greed value pairs that form a strategic hindrance space in a 3 × 2 game can be contained within a convex polygon with at most four sides—we conjecture that the number of sides of the convex polygon containing all fear and greed value pairs in an individual strategic hindrance space is at most 4(n − 2) for n > 2.

The payoffs and absolute deviation losses that produce the strategic hindrance space in Fig 4 are provided in Fig 5. Using Eqs (14) and (15)–(17) we obtain player i’s strategic hindrance space:

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Sample individual payoffs and absolute deviation losses in a 3 × 2 game.

The resulting strategic hindrance space is shown in Fig 4; the peak-to-peak payoff amplitude is A_i = 5.

https://doi.org/10.1371/journal.pone.0301394.g005

The five fear and greed value pairs in the above strategic hindrance space are all unique and fall across three 2 × 2 strategy dynamical domains—harmony (1), defection (1), and bistability (3). The center of mass is located at the average structural fear and greed coordinates, and . There are always γ(n − 1, 0) pairs along the ordinate and γ(0, n − 1) along the abscissa , with their intersection representing the player-reduced binary game where all other players play as one.

In games where players’ incentives are sparser, pre-existing beliefs of what others would do could make a player dismiss the likelihood of specific outcomes. The strategic hindrance space represents the spectrum of such beliefs, each of which a standpoint player would weigh according to a perceived chance, guiding their presumptions about the course of the game. If the standpoint player in our hypothetical 3 × 2 game example were to assign the same weight to all instances in Fig 4, the most prevalent strategy dynamics affecting their decision-making process would be structural bistability. This would involve the anticipation of aligning their strategies with at least one of the two other players. Since the remaining emergent strategy dynamics are harmony and defection, the standpoint player would prefer everyone to pursue a unanimous course of action, fully embracing either ψ or φ.

Inferring collective stability from multiple strategic hindrance spaces.

The strategic hindrance space in Fig 4 is a prognostic of a single player’s stability of rational strategic action based on their payoff structure. Suppose all other players shared similar payoff structures; in this scenario, we can anticipate the emergence of PNE coherent with the individual rational preferences coinciding in the same strategy dynamical domains. In some circumstances, we can also repeal PNE by inverting some players’ strategy dynamics, breaking compatibilities apart. For the grand game to exhibit PNE consistent with player 1’s preferences, the payoff structures of the other players must also result in a combination of harmony, bistability, and defection dynamics. Fig 6 highlights such a scenario with similar strategic hindrances, with player 2’s strategic hindrance space spanning the same three strategy dynamical domains as player 1’s, and all player 3’s fear and greed value pairs (weakly) falling under bistability. The baseline 3 × 2 game is presented in Fig 7. The PNE set in this game, , results from the intersection between the rational preferences of all three players under bistability dynamics (which already favor φ and ψ) and the strict stability under harmony and defection dynamics for players 1 and 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Strategic hindrance spaces in the sample 3 × 2 game in Fig 7.

Player 1’s payoff structure is taken from Fig 5, so their strategic hindrance space is the same as in Fig 4. Bistability dynamics prevail across the strategic hindrance spaces of all players, while the emergence of harmony and defection may add further appeal to the diagonal collective strategies ψ and φ.

https://doi.org/10.1371/journal.pone.0301394.g006

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Sample 3 × 2 game used to generate the strategic hindrance spaces in Fig 6.

Player 1’s incentives are modeled after the payoff structure in Fig 5. This game has two PNE: φ and ψ.

https://doi.org/10.1371/journal.pone.0301394.g007

In contrast with the strategic hindrance spaces in Fig 6, we can create a scenario where there is no set of type I player-reduced game observed by all players that strictly falls under the same 2 × 2 strategy dynamical domain. We can achieve this by negating player 1’s payoffs in Fig 7. The resulting game is shared in Fig 8 using an alternative 3-cub representation to highlight the directions of strict and weak payoff improvement to identify PNE, with the vertical, horizontal, and depth axes listing the individual strategies for players 1, 2, and 3, respectively. Each vertex represents a collective strategy outcome. The direction of each orthogonal edge connecting two vertices is determined by the sign of the individual deviation loss associated with it. A vertex is a PNE if it is a sink [36], meaning that all arcs connected to it are incoming [36]. There are no sinks in Fig 6; so, there are no PNE. The deviation losses and strategic hindrance spaces in this game are provided in Table 2 and visualized in Fig 9.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Modified 3 × 2 game after negating player 1’s payoffs in Fig 7, resulting in no PNE.

This 3-cube representation highlights the anticipated lack of convergence toward a collective strategy.

https://doi.org/10.1371/journal.pone.0301394.g008

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Strategic hindrance spaces in the 3 × 2 game in Fig 8.

All strategic hindrance spaces but player 1’s are the same as in Fig 6—the payoff structures of players 2 and 3 are taken from Fig 7. The coexistence dynamics that emerge for player 1 indicate a greater incentive for refusing to align with the strategies of players 2 and 3.

https://doi.org/10.1371/journal.pone.0301394.g009

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Deviation losses and strategic hindrance spaces for the 3 × 2 game in Fig 8.

All spaces are visualized in Fig 9. The results for players 2 and 3 are also shared in Fig 6.

https://doi.org/10.1371/journal.pone.0301394.t002

Negating an individual payoff structure causes observed harmony dynamics to turn into defection; it also causes bistability to turn into coexistence (and vice-versa). Harmony and defection are both dominance dynamics and thus are compatible, as described for 2 × 2 games in Section 2.3 and Eq (10)—in Section A.4 in S1 Appendix, we discuss public goods games, which have a single PNE and where the aspiration level of each player can be set to result in either pure defection or pure harmony dynamics. The same cannot be said about coexistence and bistability. In the 2 × 2 games in Eqs (11) and (12), we observe that combining positive and negative frequency dependence may result in the absence of PNE. Coexistence dynamics do not favor a unanimous course of action. Player 1’s coexistence-based preferences in Fig 9 cancel out the intersecting harmony-bistability-defection preferences of players 2 and 3 that favor φ and ψ.

It is still possible to create a 3 × 2 game where the incidence of coexistence and bistability results in the emergence of PNE. This can be done by mitigating harmony and defection. For instance, in Fig 8, making U₂(s) = 0 at s = 〈φ₁, φ₂, ψ₃〉 and s = 〈ψ₁, ψ₂, φ₃〉 makes the rescaled deviation losses and in Table 2 become zero, turning all but one of the g(3) = 5 type I player-reduced games into borderline bistability games with either F₂ = 0 or G₂ = 0. Player 2’s modified strategic hindrance is similar to that of player 3, as shown in Fig 10—unanimity games, in the following section, are characterized by the emergence of analogous borderline bistability dynamics for all players. The lower left and upper right vertices in Fig 8—respectively {〈φ₁, ψ₂, ψ₃〉 and 〈ψ₁, φ₂, φ₃〉}—weakly become sinks. These weak PNE result from the intersection of player 1’s coexistence-driven disfavoring of unanimity and players 2 and 3’s bistability-driven inclination to coordinate with other players. Here the incidence of player 1’s observed harmony and defection dynamics is negligible.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Altered strategy dynamics derived from the 3 × 2 game in Fig 8 after making U₂(s) = 0 at s = 〈φ₁, φ₂, ψ₃〉 and s = 〈ψ₁, ψ₂, φ₃〉, resulting in in Table 2.

The strategic hindrance spaces of players 2 and 3 are akin to those in a unanimity game (Fig A.5 in S1 Appendix) characterized in Section A.4 in S1 Appendix. The PNE set is .

https://doi.org/10.1371/journal.pone.0301394.g010

3.3 Application case: Integrability of an urban transit system

Here, we analyze the emergence of strategy dynamics in the strategic design for the integrability of a fictional urban transit system, highlighting the complexities that arise as the number of key design actors increases. In this example, we explore the concept of directional integrability [14], briefly discussed in Section 1.1, to assess how well the system-of-systems aligns with shared goals, considering the levels of managerial control and collaboration among its components.

The baseline 2 × 2 scenario.

Let us consider two managerially and operationally independent system design actors:

• Irene, a software developer who was contracted by the public transportation authority to implement a passenger information system; and
• Jamie, an infrastructure contractor whom the regional transportation policy agency tasked to design and maintain the traffic management system.

Irene’s primary role is to help commuters plan their routes and access real-time information on buses, trams, and subways, while Jamie’s contribution to developing and managing road infrastructure and traffic critically impacts the flow of vehicles and the safety of passengers.

Each actor can independently choose an open or closed system strategy:

• φ_i—closed system to leverage partial integrability: the system will take advantage of the capabilities of other open systems, but the latter cannot exploit the former.
• ψ_i—open system to pursue synergistic integrability: the system’s capabilities can be fully leveraged by everyone.

We consider keeping the system closed as the status quo strategy since it would not require an actor to de-constrain and expand the capabilities of their system beyond its boundaries to interact with others. In this context, pursuing synergistic integrability would be an alternative course of action—see Fig 11(a). We justify these assumptions by arguing that strategic decisions have an intended direction and are irreversible [37]. We also disregard potential halfway or “mixed” contingency plans, assuming that any decision to deviate from the status quo can be construed as a technical variation of the same alternative strategy. Thus, we focus on the incentives that affect a system actor’s decision to abandon an already in-place or taken-for-granted strategy in favor of a presumably riskier technological alternative whose potential depends on whether other actors adopt it.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11.

Increasing integrability strategically: (a) Each actor (say, actor i) decides whether to keep their system closed (status quo) or to allocate resources to making their system fully integrable for the benefit of other systems (alternative strategy); (b) The expected outcomes resulting from the interaction between two systems, i and j, as a tradeoff between the value of synergistic integrability versus the risky prospect of taking advantage of the other system’s increased openness; and (c) hypothetical assessment of system actor i’s expected outcomes from choosing to open or close their system (rows) contingent on actor j’s strategy (columns).

https://doi.org/10.1371/journal.pone.0301394.g011

Potential impact of increasing the number of actors.

Assume the involvement of a third actor named Kumar, who has been tasked to oversee safety and communications in the larger urban transit ecosystem and whose actions are critical to achieving synergistic integrability. Like Irene and Jamie, Kumar is presumed to be operationally and managerially independent in implementing an open or a closed system; however, Irene and Jamie may not be aware of Kumar’s exact level of autonomy. For example, Irene may believe that a governmental agency supersedes Kumar’s authority in whole or part and that whatever Kumar can do has already been defined. From Irene’s standpoint:

1. (a). Kumar may never open their system—a plausible assumption considering the prioritization of security, reliability, data privacy, and regulatory compliance.
2. (b). Kumar may surely open the safety and communications system for scalability, adaptiveness, and transparency.

In the context of the volunteer’s dilemma, it can be inferred that if Irene decides to maintain a closed system, it will likely result in favorable outcomes in scenario (b). However, if Kumar chooses not to pursue integrability, the risk of peril is greater in the event of (a). Notice that, in scenarios (a) and (b), it could be Jamie, not Kumar, the one presumed to have fixed their strategy. Also, it is possible that

1. (c). Jamie and Kumar align their strategies, simultaneously opening or closing their systems,

which, as in scenario (b), would make the closed system strategy riskier. If we assign the indices i, j, and k to the three actors, we can list the following five player-reduced binary games from the standpoint of actor i:

1. (1). actor j’s strategy may be uncertain, but actor k will surely maintain the status quo;
2. (2). actor j’s strategy may be uncertain, but actor k will surely choose the alternative;
3. (3). actor k’s strategy may be uncertain, but actor j will surely maintain the status quo;
4. (4). actor k’s strategy may be uncertain, but actor j will surely choose the alternative;
5. (5). both j’s and k’s strategies may be uncertain, but they will surely align their actions.

Fig 12 lists each of these games. Three of them exhibit coexistence dynamics and the remaining two exhibit defection dynamics. Specifically, using Eq. (A.7) in Section A.4 in the S1 Appendix, we obtain the strategic hindrance

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 12. Type I player-reduced binary games that would be observed by standpoint player i in a 3 × 2 volunteer’s dilemma per Eq (22).

https://doi.org/10.1371/journal.pone.0301394.g012

If we add a fourth actor, the ratio of coexistence to defection becomes 7 to 12. With five actors, it becomes 15 to 50. The incidence of defection dynamics on each player’s strategic hindrance grows at a higher rate than the incidence of coexistence dynamics as n grows; more specifically, We plot the evolution of coexistence and defection dynamics in the n-player volunteer’s dilemma in Fig 13 as the share of the total number of type I player-reduced binary games, g(n), that fall into each domain as the number of actors increases. In the context of our urban transit system example, we can expect the progress of integrability across the ecosystem to become stagnant after one actor (if any) has broken away from the status quo and implemented an open system (one of n possible PNE), assuming the actors are “economically rational.” However, if more subsystems were to be integrated, the increasing predominance of defection dynamics would lead to a decreasing likelihood of any system actor volunteering and opening their subsystems. This quantitative insight into the evolution of defection in the volunteer’s dilemma game is reminiscent of the prevalence of the diffusion of responsibility and bystander effect phenomena observed in real-world settings [38].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 13. Evolution of the strategy dynamics in n-player volunteer’s dilemma as a proportion of the number of 〈F_i, G_i〉 pairs in an individual strategic hindrance space that fall on a specific domain to the total number of type I player-reduced games, g(n).

The incidence of defection dynamics in the volunteer’s dilemma becomes more significant as n grows, while the proportion of type I player-reduced games with coexistence dynamics decreases.

https://doi.org/10.1371/journal.pone.0301394.g013

A key takeaway of this example is how crucial the efficient coordination of actions among stakeholders and systems is for optimizing resource allocation and reaching greater collective performance. For instance, if we wanted to incentivize systems-of-systems integrability in this hypothetical setting, one potential approach is to organize interconnected subsystems into small clusters to minimize the risk of coordination failure. In real-world systems-of-systems, we expect actors’ strategic tradeoffs and information to be asymmetrical and evolving, requiring the design of mechanisms to align incentives and optimize the allocation of resources. As engineering systems become more complex, managing integration and reducing incompatibility among diverse subsystems becomes increasingly challenging. Tackling these challenges will enable the management of interdependent subsystems in a harmonious way throughout each stage of the design process, ultimately leading to the creation of efficient and effective systems-of-systems that benefit society as a whole.