2.1 Utility-based methods in systems design

This section builds on a line of literature dating to early works by Simon (Reference Simon1959) that develop and apply economic methods for decision-making to a broad class of problems including engineering design. From this perspective, engineering design selects the alternative concept with highest value (under expectation) measured using von Neumann–Morgenstern utility (Hazelrigg Reference Hazelrigg1998). This approach is normative to organize and process individual preferences to support decision-making rather than descriptive to explain why certain decisions were made (Thurston Reference Thurston2001). Multi-criteria decision analysis methods including multi-attribute utility theory, analytic hierarchy process, and others help to formulate decisions for complex problems (Velasquez and Hester Reference Velasquez and Hester2013).

Applying multi-criteria decision analysis to collective systems design problems characteristically transforms each participating actor’s preferences into objective functions, defining an optimization problem. Although arguments have been made both against and in favor of this approach, most engineering design literature does not address essential uncertainty resulting from interactive effects between independent decision-makers. For instance, comparing engineering design to a social choice problem and grounded on Arrow’s (Reference Arrow1963) Impossibility Theorem, Hazelrigg (Reference Hazelrigg1997) argues that an optimal solution can only be reached if all actors share the same utility function; otherwise, any attempt to maximize aggregated individual gains is bound to result in ‘irrational’ outcomes – assuming that a ‘rational’ design maximizes every designer’s expected utility. In contrast, Scott and Antonsson (Reference Scott and Antonsson1999) state the purpose of engineering problems is to meet requirements, not individuals’ wishes, and thus engineering design exists on a continuum between single- and multi-actor problems and any disparity between system requirements and decision-makers’ preferences should be made explicit. However, in a collective systems design process with independent decision authority, actors strategically retain information and, as a result, do not have full knowledge about each other’s preferences.

Recent design research emphasizes value-centric or value-driven design methods building on rational decision-making theory to maximize system value rather than meeting requirements at minimum cost (Collopy and Hollingsworth Reference Collopy and Hollingsworth2011). A related class of methods for tradespace exploration enumerate a design space and evaluate one or more design attributes to visualize a set of alternatives (Ross et al. Reference Ross, Hastings, Warmkessel and Diller2004). These approaches generally treat risk as uncertainty or variation in value which can be analyzed with Monte Carlo sampling of a stochastic value function (Walton and Hastings Reference Walton and Hastings2004; O’Neill et al. Reference O’Neill, Yue, Nag, Grogan and de Weck2010; Daniels and Paté-Cornell Reference Daniels and Paté-Cornell2017). When applied to collective systems design, tradespace exploration results in computationally intense calculations due to combinatorial factors of a large design space coupled with interactive effects between actors. More importantly, this type of risk should not be considered as an explicit attribute to be traded during concept evaluation but only as uncertainty on other attributes (Abbas and Cadenbach Reference Abbas and Cadenbach2018). More focused analysis of strategic dynamics is necessary to understand risk in collaborative systems.

2.2 Game-theoretic methods in systems design

In an attempt to reconcile multi-criteria decision analysis methods and limitations imposed from social choice applied to engineering design, Franssen and Bucciarelli (Reference Franssen and Bucciarelli2004) demonstrate how a game-theoretic approach to collective systems design can help designers reach satisfactory outcomes without disregarding the implications of conflicting actors’ preferences. Game theory analyzes strategic decision-making among multiple interacting actors or players. In the context of engineering design, the players are the design actors – cognizant individuals, computational agents, design organizations, or indirect stakeholders with undefined strategic interests (usually modeled as players from ‘nature’). Individual utility functions, typically referred to as payoff functions, describe the players’ preferences over the possible outcomes resulting from their actions. Each player also has a set of complete contingent plans or sequences of actions, i.e. strategies, from which to choose to maximize their expected gains depending on how much information they have about the other players’ actions.

Game-theoretical approaches to engineering design frequently do not provide enough details about what aspects of the problem equate to ‘strategies’ and what other elements constitute the strategic setting of collective action or ‘game’ to be studied. An existing body of work applies game theory to engineering design by equating design alternatives to strategy spaces (Vincent Reference Vincent1983; Lewis and Mistree Reference Lewis and Mistree1997; Briceño Reference Briceño2008; Wernz and Deshmukh Reference Wernz and Deshmukh2010). In these works, the strategy set is composed of continuous functions linked to the functional attributes of the system and game-theoretical methods inform design decision-making at a low level of abstraction. Normative methods evaluate and select design decisions based on idealistic scenarios of cooperative or non-cooperative strategic equilibria (Papageorgiou et al. Reference Papageorgiou, Eres and Scanlan2016).

The majority of contributions to engineering design literature grounded in game-theoretic methods search for stable design sets under strong assumptions of rationality – namely Nash equilibria – as solutions to the multi-actor design problem. To improve outcomes designated by Nash equilibria, some works develop methods to further explore the strategy space beyond rational reaction strategy sets (Gurnani and Lewis Reference Gurnani and Lewis2008; Herrmann Reference Herrmann2010). Other works explore subgame perfect equilibria as solution concepts to game-theoretical models of engineering design (Bhatia et al. Reference Bhatia, Kannan and Bloebaum2016; Kang et al. Reference Kang, Ren, Feinberg and Papalambros2016). Finding equilibria is computationally intense and sensitive to the strategy space definition, thus the number of design alternatives to be assessed should be kept at a minimum to allow for the best use of classical game-theoretical methods. Moreover, the existence of more than one equilibrium still leaves selection of a ‘best’ option in the air and rekindles the debate about what an optimal solution means from an rational/objective perspective.

2.3 Stag hunt game and risk dominance

In contrast to existing works applying game theory to engineering design which conflate design and strategy decisions for general problems, this paper adopts a simple strategic context to evaluate dynamics for a specific class of problems related to collective systems design. Analysis of risk dominance, a concept from equilibrium selection literature, provides insights about the relationships between interacting design actors and the relative stability of collaborative systems.

The stag hunt is a canonical game theory problem that models fundamental challenges in collective decision-making (Skryms Reference Skryms2004). It follows the narrative of two hunters deciding between two alternatives to either hunt stag or hare. A stag hunt provides a desirable reward but requires joint participation of both hunters (i.e. a single stag hunter goes home hungry). A hare hunt yields a modest reward and can either be performed alone or jointly. Depending on the particular game, an independent hare hunt may be more or less rewarding than a joint hare hunt; however, both cases must be preferred to a failed stag hunt and less desirable than a successful stag hunt.

Table 1.

Stag hunt game with $u_{i}=\frac{2}{3}$

Table 1 shows a normal form payoff matrix for an example symmetric two-player stag hunt game with payoffs of 2 for a joint hare hunt, 3 for an individual hare hunt, 4 for a successful stag hunt, and 0 for a failed stag hunt. Rather than absolute wealth or resource quantities (e.g. amount of food), this paper treats payoff values as von Neumann–Morgenstern utilities that already account for behavioral factors such as loss aversion and diminishing sensitivity within each strategic context (Arrow Reference Arrow1971; Tversky and Kahneman Reference Tversky and Kahneman1992).

The strategy space ${\mathcal{S}}_{i}=\{\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D713}_{i}\}$ denotes the hare and stag strategies, respectively, for player $i$ . Inspection of selected strategy sets for two players ( $s_{1},s_{2}$ ) shows both $\unicode[STIX]{x1D719}=(\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2})$ and $\unicode[STIX]{x1D713}=(\unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2})$ are Nash equilibria because neither player has unilateral incentive to deviate away from these points. From an equilibrium perspective both stag and hare strategies are stable; however, there are clear differences in risk and reward.

Figure 1.

The expected value of strategies for player $i$ as a function of $p_{j}$ , the probability that player $j$ chooses $\unicode[STIX]{x1D713}_{j}$ for the example game in Table 1.

Harsanyi and Selten (Reference Harsanyi and Selten1988) develop theory for equilibrium selection in bipolar games based on the concept of risk dominance. Similar to how some equilibria exhibit payoff dominance (i.e. the stag equilibrium $\unicode[STIX]{x1D713}$ yields higher payoffs), risk dominance is a desirable feature that captures resistance to losses.

In specific games such as the stag hunt with two strategies and two Nash equilibria (bipolar games), further analysis compares alternative strategies from a rational (expected value maximization) perspective. Figure 1 visualizes the expected value for player $i$ as a function of the probability that player $j$ chooses strategy $\unicode[STIX]{x1D713}_{j}$ (i.e.  $p_{j}=Pr\{s_{j}=\unicode[STIX]{x1D713}_{j}\}$ ). For low values of $p_{j}$ the hare strategy $\unicode[STIX]{x1D719}_{i}$ provides the highest expected value. Similarly, for high values of $p_{j}$ the stag strategy $\unicode[STIX]{x1D713}_{i}$ provides the highest expected value. However, the two lines intersect at a point $0<u_{i}<1$ which measures the minimum probability of player $j$ choosing strategy $\unicode[STIX]{x1D713}_{j}$ for it to be rational for player $i$ to choose strategy $\unicode[STIX]{x1D713}_{i}$ .

A closed-form expression in Eq. (1) computes $u_{i}$ for any two-player bipolar game for a given payoff function $V$ .

(1) $$\begin{eqnarray}u_{i}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=\frac{V_{i}(\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D719}_{j})-V_{i}(\unicode[STIX]{x1D713}_{i},\unicode[STIX]{x1D719}_{j})}{\left(V_{i}(\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D719}_{j})-V_{i}(\unicode[STIX]{x1D713}_{i},\unicode[STIX]{x1D719}_{j})\right)-\left(V_{i}(\unicode[STIX]{x1D713}_{i},\unicode[STIX]{x1D713}_{j})-V_{i}(\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D713}_{j})\right)}.\end{eqnarray}$$

For the example in Table 1, $u_{1}=u_{2}=2/3$ . In other words, the expected value-maximizing decision for either player is to choose a stag hunt if and only if they estimate a better than two-in-three chance that their partner chooses a stag hunt.

Variations on the stag hunt game produce different values of $u_{i}$ . For example, Table 2 shows an alternative game for a scenario with a ‘trophy’ stag which increases the upside payoff from 4 to 11, lowering the threshold for economic collaboration to $u_{i}=0.2$ . Alternatively, Table 3 shows a scenario with an ‘injury’ incurred from a solitary stag hunt which decreases the downside payoff from 0 to $-2$ , raising the threshold for economic collaboration to $u_{i}=0.8$ . These scenarios show that the structure of the payoff function influences perception of collaboration and is central to the concept of risk dominance.

Table 2.

Stag hunt game with $u_{i}=0.2$

Table 3.

Stag hunt game with $u_{i}=0.8$

In the literature, $u_{i}$ is called the normalized deviation loss because its mathematical expression resembles a loss associated with deviating away from the equilibrium $\unicode[STIX]{x1D719}$ in the numerator normalized by the total losses associated with deviating away both equilibria in the denominator. Counter-intuitively, large deviation losses insulate a decision. The example in Table 2 shows the deviation loss from 11 to 3 promotes the stag strategy while the example in Table 3 shows the large deviation loss from 2 to $-2$ promotes the hare strategy.

Selten (Reference Selten1995) proposes a quantitative metric to measure risk dominance for bipolar games with linear incentives called the weighted average log measure (WALM) of risk dominance (see Appendix A for details). Linear incentives assume each player’s payoff can be expressed as a linear combination of whether other players participate in the collective strategy and is further addressed in Appendix B. Equation (2) defines the WALM of risk dominance for an $n$ -player game, where $u_{i}$ are normalized deviation losses and $w_{i}$ are influence weights based on an influence matrix $A$ which measures player interdependence (Selten Reference Selten1995).

(2) $$\begin{eqnarray}R(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\equiv \mathop{\sum }_{i=1}^{n}w_{i}(A)\ln \frac{u_{i}}{1-u_{i}}.\end{eqnarray}$$

This expression maps bipolar games to a real-number scale that measures the risk dominance of the collective strategy $\unicode[STIX]{x1D713}$ relative to the independent strategy $\unicode[STIX]{x1D719}$ . For the objective case with no knowledge of other players’ actions (equivalent to $p_{j}=0.5$ ), $R>0$ indicates $\unicode[STIX]{x1D719}$ is risk dominant while $R<0$ indicates $\unicode[STIX]{x1D713}$ is risk dominant. In all other cases with partial information leading to a probability distribution $f(p_{j})$ , $R$ provides a relative measure of risk dominance.

Figure 2.

WALM of risk dominance is the logit function of the normalized deviation loss $u_{i}$ for symmetric games with $n=2$ players.

WALM risk dominance can be simplified for games with $n=2$ players because weights are defined as $w_{1}=w_{2}=1/2$ . Furthermore, payoff symmetry with $u_{1}=u_{2}=u_{i}$ further reduces the expression to Eq. (3) which is simply the logit function of $u_{i}$ visualized in Figure 2.

(3) $$\begin{eqnarray}R(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=\ln \frac{u_{i}}{1-u_{i}}=\ln \left(\frac{V_{i}(\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D719}_{j})-V_{i}(\unicode[STIX]{x1D713}_{i},\unicode[STIX]{x1D719}_{j})}{V_{i}(\unicode[STIX]{x1D713}_{i},\unicode[STIX]{x1D713}_{j})-V_{i}(\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D713}_{j})}\right).\end{eqnarray}$$

The stag hunt games in Tables 1–3 have $R_{1}=\ln 2\approx 0.69$ , $R_{2}=\ln 0.25\approx -1.39$ , and $R_{3}=\ln 4\approx 1.39$ . In Table 2, $R_{2}<0$ indicates the collective strategy $\unicode[STIX]{x1D713}$ is risk dominant. In Table 1 and Table 3, $R_{3}>R_{1}>0$ indicates the independent strategy $\unicode[STIX]{x1D719}$ is risk dominant and more strongly so in Table 3 compared to Table 1. Risk dominance is normative for strategy selection in non-cooperative cases; however, in other cases it is only a relative measure of strategic dynamics across games.

While direct analysis of individual incentives in Figure 1 provides an intuitive explanation of strategic dynamics in two-player bipolar games, the more general formulation for WALM of risk dominance detailed in Appendix A handles asymmetric games with $n\geqslant 2$ players where players may express different or conflicting normalized deviation losses and pairwise interactions.

3.1 Strategic design games

Game theory is a strategic analysis method that works at a high level of abstraction. In the context of engineering design, strategic decision-making best corresponds to architecture selection in conceptual design rather than more detailed design decisions in preliminary design. This section defines the concept of a strategic design game as a multi-actor value model for engineering decision-making to permit strategic analysis (Grogan et al. Reference Grogan, Ho, Golkar and de Weck2018).

Table 4.

Stag hunt design utilities

Table 5.

Stag hunt game with $u_{i}=0.5$

A strategic design game distinguishes between two levels of decisions: strategy decisions $s_{i}\in {\mathcal{S}}_{i}$ govern collective behavior among players and design decisions $d_{i}\in {\mathcal{D}}_{i}$ specify system configurations. A corresponding multi-actor value function $[V_{i}^{s_{1},\ldots ,s_{n}}(d_{1},\ldots ,d_{n})]$ for $n$ players in Eq. (4) maps design and strategy decisions to values (utilities) for each player.

(4) $$\begin{eqnarray}V:\mathop{\prod }_{i=1}^{n}{\mathcal{D}}_{i}\times {\mathcal{S}}_{i}\rightarrow \mathbb{R}^{n}.\end{eqnarray}$$

While the design spaces ${\mathcal{D}}_{i}$ may be large or unbounded and unique to each player, the strategy space ${\mathcal{S}}_{i}=\{\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D713}_{i}\}$ is limited to two options: choosing between independent ( $\unicode[STIX]{x1D719}_{i}$ ) or collective ( $\unicode[STIX]{x1D713}_{i}$ ) action. Within each strategy space, design decisions are evaluated based on the context of the governing strategy.

To illustrate this concept, reconsider the stag hunt game from Table 1 with design variables to choose the hunting weapon from the symmetric design space ${\mathcal{D}}=\{\text{Atlatl},\text{Bow},\text{Club},\text{Dog}\}$ . A multi-actor value model in Table 4 evaluates each design alternative in possible strategic contexts. Within fixed equilibrium contexts the best hare-hunting design is $\text{Dog}$ with utility 2 and the best stag-hunting design is $\text{Atlatl}$ with utility 4. However, as previously investigated in Table 1, this combination requires a probability greater than $u_{i}=2/3$ to pursue a stag hunt. In cases with unreliable partners, the stag-hunting design $\text{Bow}$ in Table 5 may be more desirable because it only requires a probability greater than $u_{i}=0.5$ to pursue a stag hunt, although it only provides utility 3.5 if successful.

Applied to engineering design, the independent strategy $\unicode[STIX]{x1D719}_{i}$ corresponds to systems designed and operated with few external dependencies. The collective strategy $\unicode[STIX]{x1D713}_{i}$ represents potential performance gains from collaborative systems having numerous interdependencies which operate at risk of degraded performance due to coordination failures. The resulting strategic design game resembles a binary game with zero, one, or two pure strategy Nash equilibria. Games with zero equilibria have no stable strategy sets, limiting the use of normative analysis. Games with one equilibrium exhibit a dominant strategy and do not benefit from further analysis. Therefore, only cases with two equilibria (i.e. bipolar games) benefit from and are valid to measure risk dominance.

More detailed analysis of a strategic design game benefits from two simplifying assumptions in Eq. (5) and shown in Table 6 for a normal form strategic design game with $n=3$ players.

(5) $$\begin{eqnarray}V_{i}^{s_{1},\ldots ,s_{n}}(d_{1},\ldots ,d_{n})=\left\{\begin{array}{@{}ll@{}}V_{i}^{\unicode[STIX]{x1D719}}(d_{i})\quad & \text{if}\,s_{i}=\unicode[STIX]{x1D719}_{i}\\ V_{i}^{\unicode[STIX]{x1D713}}(d_{k}:s_{k}=\unicode[STIX]{x1D713}_{k})\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$

These assumptions limit interaction effects between participants inside and outside a collective strategy and are not strictly required for analysis but help to communicate the method and results using simplified notation.

The first simplification approximates the multi-actor value function by a single-actor value function $V_{i}^{\unicode[STIX]{x1D719}}$ when player $i$ chooses an independent strategy $\unicode[STIX]{x1D719}_{i}$ . This assumes no strong interaction effects between independent players and others and allows local design optimization in Eq. (6).

(6) $$\begin{eqnarray}{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}=\max _{d\in {\mathcal{D}}_{i}}V_{i}^{\unicode[STIX]{x1D719}}(d).\end{eqnarray}$$

The second simplification aggregates all candidate designs for players participating in a collective strategy $\unicode[STIX]{x1D713}$ including player $i$ . For the special case where no other players select the collective strategy (i.e.  $s_{j}=\unicode[STIX]{x1D719}_{j}\;\forall j\neq i$ ), a single-actor value function $V_{i}^{\unicode[STIX]{x1D713}}(d_{i})$ replaces the multi-actor one. This formulation emphasizes interaction effects among participants in a collective strategy but assumes no strong interaction effects with players outside.

Table 6.

Strategic design game for three players

3.2 Strategic risk dominance

This section uses the strategic design game concept to explain and measure risk dominance in collaborative systems design. Risk dominance is only meaningful in bipolar games which requires a specific ordering of value quantities such that $V_{i}^{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})>{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}>V_{i}^{\unicode[STIX]{x1D713}}(d_{i})\;\forall i$ . This is a reasonable requirement because it represents the cases of most interest where the collaborative system has a higher upside potential but also downside risk to an independent alternative.

As introduced in Section 2.3 and detailed in Appendix A, risk dominance first depends on normalized deviation losses. Equation (7) shows the normalized deviation loss for player $i$ by substituting the strategic design game notation into Eq. (1).

(7) $$\begin{eqnarray}\displaystyle u_{i} & = & \displaystyle \frac{{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})}{\left({\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})\right)+\left(V_{i}^{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})-{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}\right)}\nonumber\\ \displaystyle & = & \displaystyle \frac{{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})}{V_{i}^{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})}.\end{eqnarray}$$

The simplest form of strategic risk dominance assumes symmetric design spaces and utility functions for all players. Symmetry yields equal weighting factors $w_{i}$ , producing the risk dominance measure in Eq. (8) as a function of the collaborative design $d$ (note: all value subscripts dropped due to symmetry).

(8) $$\begin{eqnarray}R_{\unicode[STIX]{x1D713}}(d)=\ln \left(\frac{{\mathcal{V}}^{\unicode[STIX]{x1D719}}-V^{\unicode[STIX]{x1D713}}(d)}{V^{\unicode[STIX]{x1D713}}(d,\ldots ,d)-{\mathcal{V}}^{\unicode[STIX]{x1D719}}}\right).\end{eqnarray}$$

Symmetric risk dominance requires three function evaluations to quantify the upside value of a successful collective strategy $V^{\unicode[STIX]{x1D713}}(d,\ldots ,d)$ , downside value of a failed collective strategy $V^{\unicode[STIX]{x1D713}}(d)$ , and the value of the independent alternative ${\mathcal{V}}^{\unicode[STIX]{x1D719}}$ .

More general forms of strategic risk dominance assume asymmetric design spaces or utility functions. As detailed in Appendix A, influence elements $a_{ij}$ measure the dependence between players $i$ and $j$ . Equation (9) shows the influence elements $a_{ij}$ between players $i$ and $j$ , assuming linear incentives, by substituting the notation for strategic design games into Eq. (27).

(9) $$\begin{eqnarray}\displaystyle a_{ij} & = & \displaystyle \frac{\left({\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})\right)-\left({\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}-V_{i}^{\unicode[STIX]{x1D713}}(d_{i},d_{j})\right)}{\left({\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})\right)+\left(V_{i}^{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})-{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}\right)}\nonumber\\ \displaystyle & = & \displaystyle \frac{V_{i}^{\unicode[STIX]{x1D713}}(d_{i},d_{j})-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})}{V_{i}^{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})}.\end{eqnarray}$$

Linear incentives enforce a constraint that all row sums total one, i.e.  $\sum _{j\neq i}a_{ij}=1\,\forall \,i$ . In cases with nonlinear incentives for $n=3$ players, as in Table 6, Eq. (10) expresses linearized influence elements $\bar{a}_{ij}$ between players $i$ and $j$ by substituting strategic design game notation into Eq. (2) (see Appendix B for details).

(10) $$\begin{eqnarray}\bar{a}_{ij}=\frac{1}{2}\left(1+a_{ij}-a_{ik}\right)=\frac{1}{2}\left(1+\frac{V_{i}^{\unicode[STIX]{x1D713}}(d_{i},d_{j})-V_{i}^{\unicode[STIX]{x1D713}}(d_{i},d_{k})}{V_{i}^{\unicode[STIX]{x1D713}}(d_{1},d_{2},d_{3})-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})}\right).\end{eqnarray}$$

After computing elements of the influence matrix $A$ using linear incentives $a_{ij}$ or the linear approximation $\bar{a}_{ij}$ , subsequent analysis finds weighting factors $w_{i}$ to measure player importance. As detailed in Appendix A, weighting factors are the eigenvector corresponding to the unit eigenvalue of the influence matrix $A$ .

The risk dominance measure in Eq. (11) computes the WALM in Eq. (2) as a function of collaborative designs $(d_{1},\ldots ,d_{n})$ .

(11) $$\begin{eqnarray}R_{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})=\mathop{\sum }_{i=1}^{n}w_{i}(A)\ln \left(\frac{{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}-V_{i}^{\unicode[STIX]{x1D713}}(d_{i})}{V_{i}^{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})-{\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}}\right).\end{eqnarray}$$

In general, asymmetric risk dominance requires $1+\sum _{k=1}^{n}\binom{n}{k}=2^{n}$ multi-actor value function evaluations to consider all possible combinations of players joining the collective strategy (required to compute $\bar{a}_{ij}$ terms) plus the independent alternative. While not burdensome for small games, this combinatorial factor may limit the use of similar methods for large games with many decision-makers.

3.3 Assumptions and limitations

This work includes several key assumptions and limitations which must be discussed. First, measuring risk dominance assumes a strategic decision-making process structured as a bipolar game. The strategic design game is an abstraction of the design process where, in reality, lower-level design and higher-level strategy decisions are coupled and iterative. Results from this work thus provide a baseline result which must be considered in the context of a specific design problem. While a limiting constraint, bipolar games are interesting cases to study because they present a fundamental tradeoff between risk and reward.

Second, this work represents an analytical and rational/objective method to measure strategic risk dominance which is both a significant limitation and a significant strength. The strategic-level analysis only aims to maximize expected value – although utility functions to compute payoff values can incorporate risk attitudes. No subjective information is required to determine or assess the likelihood of other players’ actions nor are there any elements of cooperative game theory to enforce contracts or share or divide benefits (although $R$ is closely related to the Nash product and $\bar{a}_{ij}$ terms resemble Shapley value). If additional subjective information were available, a more thorough analysis leveraging Bayesian games could be performed, as pioneered by Harsanyi (Reference Harsanyi1967). In light of this limitation, relative values of the WALM risk dominance measure (e.g. during an architecture trade study) are more important than absolute values.

Third, the existing equilibrium selection theory imposes a few restrictions on the types of problems modeled. As discussed in Appendix A, it assumes linear incentives which are unlikely to hold in most engineering applications due to economies of scale; however, the linear approximation methods and error analysis introduced mitigate some of this concern. Developing utility functions to quantify payoff values while accommodating behavioral factors such as risk attitudes remains a practical challenge which is out of the scope of this paper.

Two other notational limitations can be relaxed for further analysis. The risk metric in Eq. (11) only assesses one set of collaborative architectures ( $d_{1},\ldots ,d_{n}$ ) under one collective strategy ( $\unicode[STIX]{x1D713}$ ) relative to the independent baseline. However, as a real number, $R$ can compare multiple design candidates relative to the same independent baseline (e.g.  $R_{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})$ vs. $R_{\unicode[STIX]{x1D713}}(d_{1}^{\prime },\ldots ,d_{n}^{\prime })$ ) to guide the design search process. Similarly, $R$ can compare multiple collective strategy candidates for a fixed set of federated alternatives (e.g.  $R_{\unicode[STIX]{x1D713}}(d_{1},\ldots ,d_{n})$ vs. $R_{\unicode[STIX]{x1D713}^{\prime }}(d_{1},\ldots ,d_{n})$ ) to guide the strategy formulation process.

4.1 Multi-actor value model and design scenarios

This section develops an application case based on a stylized model of federated Earth-observing space systems paired with an existing simulation model. Orbital Federates Simulation – Python (OFSPY) (Grogan Reference Grogan2019) acts as a multi-actor value function by mapping design and strategy sets (inputs) to net present value earned by each player over a simulated system lifetime (outputs). The model includes stochastic features to capture operational uncertainty such that results must be sampled using Monte Carlo methods. While many model details are clearly fictional, the underlying model was developed to have structural and process isomorphic features to help understand strategic player behavior for space systems (Grogan and de Weck Reference Grogan and de Weck2015).

This application case only evaluates how to model collaborative design scenarios as a strategic design game and compute measures of risk dominance. The implementation details of the multi-actor value model are outside the scope of this paper; however, Appendix C provides more details for replication.

The design scenario considers $n=3$ players who operate space systems to collect and downlink data to satisfy demands and earn revenue. Figure 3 illustrates designs selected from a large combinatorial design space for a baseline (independent) strategic context and two federated alternatives. The independent case includes small standalone observing spacecraft for players 2 and 3 who specialize in synthetic aperture radar (SAR) and visual light (VIS) sensors, respectively. Player 1 does not participate in the baseline system. Federated designs consider an opportunistic data exchange policy with a fixed price for inter-satellite link (ISL) and space-to-ground (SGL) services among players. Federated scenario A includes participation by player 1 with a data relay spacecraft and SGL receiver and ISL adoption among all three players. Federated scenario B eliminates the ISL technology option and establishes an independent observing spacecraft for player 1 with the SGL receiver.

Figure 3.

Initial ground station and satellite locations in a two-dimensional space numbered by player for (a) baseline, (b) scenario A, and (c) scenario B. Dotted lines indicate SGL or ISL for the initial conditions.

Table 7 shows the expected net present value for each strategic context using a discount rate of 2 $\%$ per turn evaluated using 1000 seeded runs of the multi-actor value function. For clarity in presentation, players are assumed to be risk neutral such that expected net present value is equivalent to utility; however, alternative assumptions of risk attitudes would modify the resulting payoff values following a nonlinear utility curve. In practice, sensitivity analyses may help understand how unknown behavioral quantities such as risk attitudes influence results.

Table 7.

Orbital federates design scenarios

Table 8.

Strategic design game for scenario A

4.2 Risk dominance analysis of scenario A

Table 8 populates a strategic design game for federated scenario A which constitutes a bipolar game with asymmetric players. Using Eq. (7), the normalized deviation losses are

(12) $$\begin{eqnarray}u=\left[\begin{array}{@{}c@{}}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]=\left[\begin{array}{@{}c@{}}0.917\\ 0.378\\ 0.365\end{array}\right].\end{eqnarray}$$

Using Eq. (10), the linearized influence matrix is

(13) $$\begin{eqnarray}A=\left[\begin{array}{@{}ccc@{}}0 & \bar{a}_{12} & \bar{a}_{13}\\ \bar{a}_{21} & 0 & \bar{a}_{23}\\ \bar{a}_{31} & \bar{a}_{32} & 0\end{array}\right]=\left[\begin{array}{@{}ccc@{}}0 & 0.533 & 0.467\\ 0.876 & 0 & 0.124\\ 0.784 & 0.216 & 0\end{array}\right].\end{eqnarray}$$

Incentive function linearization error analysis using Eq. (9) shows small errors for all three players: $\unicode[STIX]{x1D700}=\left[0.047,0.010,0.024\right]$ .

Eigenvector analysis of the transposed influence matrix $A^{\intercal }$ yields the eigenvector (rescaled to unit norm)

(14) $$\begin{eqnarray}w(A)=\left[\begin{array}{@{}c@{}}w_{1}(A)\\ w_{2}(A)\\ w_{3}(A)\end{array}\right]=\left[\begin{array}{@{}c@{}}0.455\\ 0.296\\ 0.249\end{array}\right]\end{eqnarray}$$

corresponding to the unit eigenvalue. Computing WALM as

(15) $$\begin{eqnarray}R_{\unicode[STIX]{x1D713}}(d_{1A},d_{2A},d_{3A})=\mathop{\sum }_{i=1}^{3}w_{i}(A)\ln \frac{u_{i}}{1-u_{i}}=0.805\end{eqnarray}$$

shows the independent strategy $\unicode[STIX]{x1D719}$ to be risk dominant.

Normalized deviation losses show the independent strategy is preferred for player 1 in all but high probabilities of collaboration ( $u_{1}=0.917$ ) because of large downside losses incurred if no other players choose the collective strategy. This risk fundamentally arises because player 1 has no independent source of revenue to recover the high cost of the federated design. However, the collective strategy has a lower threshold of collaboration for players 2 and 3 ( $u_{2}=0.378,u_{3}=0.365$ ) because large upside gains realized from shared data services overcome the additional cost of larger spacecraft and extra modules.

Weighting factors identify player 1 as the most influential ( $w_{1}=0.455$ ) which can be explained by their central role in both providing shared ISL relay and SGL downlink services via the spacecraft and ground station, respectively. Smaller but similar weights for players 2 and 3 ( $w_{2}=0.296,w_{3}=0.249$ ) reflect their similarity in operational mission.

Given player 1’s aversion to the collective strategy and strong influence, the independent strategy is risk dominant in this design scenario. In particular, the collective strategy is prone to failure due to disengagement by player 1. A visualization of value surfaces in Figure 4 emphasizes the disparity between players 1 compared to 2 and 3 with respect to overall stability of the collective strategy.

Figure 4.

Value surfaces for independent strategy $\unicode[STIX]{x1D719}_{i}$ versus collective strategy $\unicode[STIX]{x1D713}_{i}$ under federated scenario A. The independent strategy $\unicode[STIX]{x1D719}$ is more dominant for player 1 while the collective strategy $\unicode[STIX]{x1D713}$ is more dominant for players 2 and 3.

Table 9.

Strategic design game for scenario B

4.3 Risk dominance analysis of scenario B

Table 9 populates a strategic design game for the federated scenario B which constitutes a bipolar game with asymmetric players. Using Eq. (7), the normalized deviation losses are

(16) $$\begin{eqnarray}u=\left[\begin{array}{@{}c@{}}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]=\left[\begin{array}{@{}c@{}}0.277\\ 0.568\\ 0.418\end{array}\right].\end{eqnarray}$$

Using in Eq. (10), the linearized influence matrix is

(17) $$\begin{eqnarray}A=\left[\begin{array}{@{}ccc@{}}0 & \bar{a}_{12} & \bar{a}_{13}\\ \bar{a}_{21} & 0 & \bar{a}_{23}\\ \bar{a}_{31} & \bar{a}_{32} & 0\end{array}\right]=\left[\begin{array}{@{}ccc@{}}0 & 0.566 & 0.434\\ 0.990 & 0 & 0.010\\ 0.962 & 0.038 & 0\end{array}\right].\end{eqnarray}$$

Incentive function linearization error analysis using Eq. (9) shows small errors for all three players: $\unicode[STIX]{x1D700}=\left[0.034,0.005,0.030\right]$ .

Eigenvector analysis of the transposed influence matrix $A^{\intercal }$ yields the eigenvector (rescaled to unit norm)

(18) $$\begin{eqnarray}w(A)=\left[\begin{array}{@{}c@{}}w_{1}(A)\\ w_{2}(A)\\ w_{3}(A)\end{array}\right]=\left[\begin{array}{@{}c@{}}0.494\\ 0.288\\ 0.217\end{array}\right]\end{eqnarray}$$

corresponding to the unit eigenvalue. Computing WALM as

(19) $$\begin{eqnarray}R_{\unicode[STIX]{x1D713}}(d_{1B},d_{2B},d_{3B})=\mathop{\sum }_{i=1}^{3}w_{i}(A)\ln \frac{u_{i}}{1-u_{i}}=-0.466\end{eqnarray}$$

shows the collective strategy $\unicode[STIX]{x1D713}$ is risk dominant.

Normalized deviation losses show the collective strategy is preferred for player 1 for a wide range of probabilities of collaboration ( $u_{1}=0.277$ ). The threshold for collaboration is higher for players 2 and 3 ( $u_{2}=0.568,u_{3}=0.418$ ) compared to scenario A. The goal of reducing upfront costs and providing an independent source of revenue successfully changed the strategic risk posture of player 1. However, the loss of ISL relay services and associated revenue disincentivize the collective strategy for players 2 and 3.

Weighting factors still identify player 1 as the most influential ( $w_{1}=0.494$ ) and more influential compared to scenario A. Player 1 retains a key role in providing downlink services and, without ISL services among players 2 and 3, takes an even stronger role because other players lack the relay components to interact with each other directly. Players 2 and 3 retain similar weights ( $w_{2}=0.288,w_{3}=0.217$ ) though both are less influential than in scenario A.

Combining these factors, the collective strategy is risk dominant in this design scenario. This result indicates scenario B has preferable strategic dynamics to scenario A at the cost of slightly lower value. Notably, in the event that player 2 disengages, players 1 and 3 still enjoy moderate returns from the collective strategy. A visualization of value surfaces in Figure 5 shows a clear difference for player 1 compared to scenario A in Figure 4.

Figure 5.

Value surfaces for independent strategy $\unicode[STIX]{x1D719}_{i}$ versus collective strategy $\unicode[STIX]{x1D713}_{i}$ under federated scenario B. The independent strategy $\unicode[STIX]{x1D719}$ is dominant for player 2 while the collective strategy $\unicode[STIX]{x1D713}$ is dominant for players 1 and 3.

4.4 Comparative analysis of results

This case illustrates how high-value (but also high-risk) designs emerge from optimization-oriented activities and how analysis of risk dominance has the potential to mitigate strategic instabilities by selecting more conservative alternatives. If successful, scenario A provides superior value for all three players by taking advantage of new technology and operational concepts. However, focusing on maximizing upside potential can yield unstable design solutions more susceptible to coordination failures. Player 1 experiences a significantly higher threshold for collaboration than others under scenario A and is most likely to disengage from a collective strategy.

Scenario B reduces the level of technological ambition and establishes an independent value source for all players. While its potential payoffs are smaller than A, scenario B exhibits superior strategic stability and is robust to disengagement by player 2, the most likely to disengage from a collective strategy. These results echo Maier’s principles for system-of-systems architecting emphasizing stable intermediate forms and ensuring cooperating among all actors (Maier Reference Maier1998). Risk dominance helps assess designs for a balance between value maximization if successful and risk minimization for coordination failures.

Although not considered in this analysis for clarity in presentation, incorporating risk attitudes would influence the absolute (but not relative) interpretation of risk dominance across scenarios A and B. For example, an exponential utility function of the form $U(c)=(1-e^{-\unicode[STIX]{x1D6FC}c})/\unicode[STIX]{x1D6FC}$ for consumption level $c$ , equivalent to expected net present value in this example, assumes constant risk aversion for $\unicode[STIX]{x1D6FC}>0$ (Arrow Reference Arrow1971). Applying this transformation to the payoff values in Table 7 penalizes the high-value (but uncertain) collaborative outcomes and increases risk dominance measures across both scenarios. More detailed analysis would benefit from specific knowledge about risk attitudes on behalf of individual players.

Alternative probabilistic analysis methods may evaluate expected value and variance under uncertain strategy selections. For example, defining $p_{i}$ as the probability player $i$ deviates from $\unicode[STIX]{x1D719}$ to $\unicode[STIX]{x1D713}$ , the value function $V_{i}$ is

(20) $$\begin{eqnarray}\displaystyle V_{i}(p_{i},p_{j},p_{k}) & = & \displaystyle {\mathcal{V}}_{i}^{\unicode[STIX]{x1D719}}(1-p_{i})+V_{i}^{\unicode[STIX]{x1D713}}(d_{i})p_{i}(1-p_{j})(1-p_{k})\nonumber\\ \displaystyle & & \displaystyle +\,V_{i}^{\unicode[STIX]{x1D713}}(d_{i},d_{k})p_{i}(1-p_{j})p_{k}+V_{i}^{\unicode[STIX]{x1D713}}(d_{i},d_{j})p_{i}p_{j}(1-p_{k})\nonumber\\ \displaystyle & & \displaystyle +\,V_{i}^{\unicode[STIX]{x1D713}}(d_{i},d_{j},d_{k})p_{i}p_{j}p_{k}.\end{eqnarray}$$

Assuming $p_{i}$ are independent and identically distributed with $p_{i}\sim \text{uniform}(0,1)$ , Table 10 reports expected value $E\left[V_{i}\right]$ and variance $\text{Var}(V_{i})$ for scenarios A and B. The analysis concurs that player 1 in scenario A and player 2 in scenario B observe negative expected value and scenario B overall decreases variance for all players. However, this analysis: 1) does not provide any tradeoff between expected value and variance, 2) only provides a relative comparison between the two cases without a scalar quantitative metric of stability, and 3) does not provide further insights for the interdependency or influence between or among players.

Table 10.

Comparative analysis of expected value and variance