## Appendix A. Detailed WALM formulation

This section summarizes key results from Selten (1995) to formulate and explain the weighted average log measure (WALM) of risk dominance. It does not produce any new results but introduces some more convenient notation and insights. Please refer to the original article for axioms and proofs.

Binary games have a strategy space with two alternatives
${\mathcal{S}}_{i}=\{\unicode[STIX]{x1D719}_{i},\unicode[STIX]{x1D713}_{i}\}$
. Bipolar games are a subclass of binary games with two Nash equilibria defined by shared strategies among all players
$\unicode[STIX]{x1D719}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{n})$
and
$\unicode[STIX]{x1D713}=(\unicode[STIX]{x1D713}_{1},\ldots ,\unicode[STIX]{x1D713}_{n})$
. Note that, for example,
$\unicode[STIX]{x1D719}$
denotes the strategy set and
$\unicode[STIX]{x1D719}_{i}$
denotes the strategy selected by player
$i$
. Notation with negative subscripts on strategy sets denotes non-participation, for example,
$\unicode[STIX]{x1D719}_{-1}=(\unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D719}_{2},\ldots ,\unicode[STIX]{x1D719}_{n})$
. Without loss of generality, this section labels strategy sets such that
$\unicode[STIX]{x1D713}$
is payoff dominant as the collective strategy.

For greater generality, WALM of risk dominance is defined in terms of a *biform* that describes the essential dynamics of a game rather than the direct payoff or utility function. The biform is given by a vector of normalized deviation losses
$u=u(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=\left[u_{i}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\right]$
and an influence matrix
$A=A(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=\left[a_{ij}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})\right]$
capturing interdependencies between players.
^{1}
Together, these factors attribute potential losses to global and local deviations from a baseline strategy.

The most intuitive explanation of risk dominance starts by formulating an incentive function
$D_{i}$
for player
$i$
to choose
$\unicode[STIX]{x1D719}_{i}$
over
$\unicode[STIX]{x1D713}_{i}$
. Player
$i$
prefers
$\unicode[STIX]{x1D719}_{i}$
for
$D_{i}>0$
and prefers
$\unicode[STIX]{x1D713}_{i}$
for
$D_{i}<0$
. An expected value expression expanded in Eq. (21) describes the incentive function from a global perspective as a function of
$p$
, the probability that *all* other players choose
$\unicode[STIX]{x1D713}$
.

Equation (22) defines deviation loss
$L_{i}$
as a function of a strategy set
$\unicode[STIX]{x1D709}\in \{\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}\}$
up to a constant of proportionality.

The deviation loss captures sensitivity to deviating away from a stable strategy set through one’s own actions; however, for the purpose of this formulation consider it an algebraic expression only. The normalized deviation loss
$u_{i}$
in Eq. (23) transforms
$L_{i}$
to a unit scale.

Returning to the global incentive function, Eq. (24) normalizes both sides and substitutes expressions for
$L_{i}$
and
$u_{i}$
to achieve a simplified incentive function.

In other words, player
$i$
prefers
$\unicode[STIX]{x1D719}_{i}$
for
$D_{i}>0\;\Longleftrightarrow \;p<u_{i}$
and prefers
$\unicode[STIX]{x1D713}_{i}$
for
$D_{i}<0\;\Longleftrightarrow \;p>u_{i}$
. The normalized deviation loss
$u_{i}$
(also referred to as the diagonal probability
$\unicode[STIX]{x1D70B}_{i}$
in literature) marks the intersection between the lines in Figure 6 where
$D_{i}(u_{i})=0$
(i.e. player
$i$
is indifferent about which strategy to select).

A more detailed incentive function can be written from a local perspective specific to each players’ strategy to capture interaction effects and interdependencies. An expected value expression expanded in Eq. (25) for a game with
$n=3$
players describes the incentive function as a function of
$p_{j}$
and
$p_{k}$
, the probability that players
$i$
and
$j$
choose
$\unicode[STIX]{x1D713}_{i}$
and
$\unicode[STIX]{x1D713}_{k}$
, respectively.

Equation (26) defines pairwise deviation loss
$L_{ij}$
as a function of a strategy set
$\unicode[STIX]{x1D709}\in \{\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}\}$
up to a constant of proportionality.

Similar to deviation loss, pairwise deviation loss captures sensitivity to deviating away from a stable strategy set through pairwise actions but for the purpose of this formulation consider it an algebraic expression only. Influence elements
$a_{ij}$
in Eq. (27) normalize pairwise deviation losses
$L_{ij}$
to a common scale with
$u_{i}$
.

Returning to the local incentive function, normalizing both sides yields the simplified form in Eq. (28).

Under the assumption of linear incentives,
$1-a_{ij}-a_{ik}=0$
such that
$D=u-Ap$
which is the general result applicable to games with any number of players.

Finally, influence weights measure the overall importance of one player on others’ stability. Weights
$w=\left[w_{i}(A)\right]$
are defined implicitly by properties in Eq. (29) based on the influence matrix
$A=\left[a_{ij}\right]$
.

Weights are interpreted as the eigenvector (rescaled to unit norm) of
$A^{\intercal }$
corresponding to the unit eigenvalue, guaranteed to exist by the assumption of linear incentives which forces the row sum of
$A$
to unity for all rows. Note that weights are equivalent to the limiting stochastic distribution for the Markov chain with state transition probabilities
$a_{ij}$
.

## Appendix B. Approximation to linear incentives

Selten’s work focuses on games with linear incentives which allow pairwise interactions to be quantified without third party effects (e.g. the effect of player
$j$
on player
$i$
is not a function of player
$k$
). This simplifying assumption, similar to a first order approximation, greatly reduces complexity for a narrow class of problems but cannot directly represent increasing or decreasing returns to scale (i.e. network effects) common in engineering applications. Furthermore, linear incentives is a critical assumption to find weighting factors which require a unit eigenvalue of the influence matrix
$A$
. Although there may be extensions of the influence matrix
$A$
to higher dimensions (e.g. tensors), there is currently no such existing theory. Thus, this section introduces a novel linear approximation for greater applicability to design problems with nonlinear incentives.

Linear incentives can be visualized as planar value surfaces in Figure 7 for a game with
$n=3$
players as a function of
$p_{j}$
and
$p_{k}$
, the probability players
$j$
and
$k$
choose strategy
$\unicode[STIX]{x1D713}$
over
$\unicode[STIX]{x1D719}$
, respectively. The incentive function
$D_{i}(p_{j},p_{k})$
is the difference between the two planes. The intersection between the two planes (black line) traces the indifference curve where player
$i$
does not prefer either strategy, similar to
$u_{i}$
for
$n=2$
players. Games with nonlinear incentives visualized in Figure 7 for an exaggerated case include interaction terms with third parties and yield non-planar value surfaces and nonlinear indifference curves.

Consider the simplest possible game with nonlinear incentives with
$n=3$
players and incentive function in Eq. (28). Linear incentives require
$a_{ij}+a_{ik}=1$
to eliminate the interaction term between
$p_{j}$
and
$p_{k}$
. A linearized incentive function in Eq. (1) proposes modified
$a_{ij}$
terms such that
$\sum _{j=1}^{n}\bar{a}_{ij}=1\;\forall \;i$
to satisfy the linear incentives condition.

Preserving influence elements as a coefficient for the effect of player
$j$
’s probability of choosing
$\unicode[STIX]{x1D713}_{j}$
on player
$i$
’s incentive to choose
$\unicode[STIX]{x1D719}_{i}$
(specifically,
$-\unicode[STIX]{x2202}D_{i}/\unicode[STIX]{x2202}p_{j}$
), Eq. (2) defines the linearized influence element
$\bar{a}_{ji}$
as the expected value of the partial derivative of the incentive function with respect to
$p_{j}$
.

For more general games with
$n>3$
players, linearizing incentive functions using this approximation becomes a combinatorial problem based on
${\mathcal{K}}_{ij}$
, the power set
${\mathcal{P}}$
of third parties (i.e. set of all subsets of players except
$i$
and
$j$
and including the empty set) in Eq. (3) with cardinality
$|{\mathcal{K}}_{ij}|$
in Eq. (4) given by the binomial theorem.

Revised notation in Eq. (5) defines combinatorial deviation losses between player
$i$
and a set of players
$\mathbf{k}$
as a function of a strategy set
$\unicode[STIX]{x1D709}\in \{\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}\}$
up to a constant of proportionality.

Note that this expression simplifies to previously established forms of
$L_{i\{\}}(\unicode[STIX]{x1D709})=L_{i}(\unicode[STIX]{x1D709})$
in Eq. (22) for
$\mathbf{k}=\{\}$
and
$L_{i\{j\}}(\unicode[STIX]{x1D709})=L_{ij}(\unicode[STIX]{x1D709})$
in Eq. (26) for
$\mathbf{k}=\{j\}$
. Using this notation, Eq. (6) states a conjecture for linearized influence elements.

While proof of this conjecture is not available, the result above has been manually verified for
$n=3$
(see Eq. (2) recognizing that
$L_{ij}(\unicode[STIX]{x1D719})=-L_{ik}(\unicode[STIX]{x1D713})$
) and
$n=4$
cases.

Linearizing influence elements introduces errors into the risk dominance analysis. Error manifests as differences between the incentive function
$D_{i}(p_{j},p_{k})$
and its linearized form in Eq. (7) for games with
$n=3$
players.

For example, Figure 8 visualizes contours of player
$i$
’s incentive function for a notional symmetric
$n=3$
game with value function

influence elements
$a_{ij}=0.25$
, and linearized influence elements
$\bar{a}_{ij}=0.5$
for (a) initially (highly) nonlinear incentives, (b) linearized incentives following the recommended method, and (c) the subsequent absolute difference
$\unicode[STIX]{x1D6FF}_{i}$
.

Equation (9) derives a simple error metric
$\unicode[STIX]{x1D700}_{i}$
for cases with
$n=3$
players that measures the average absolute difference in incentive value.

Note that
$D_{i}(p_{j},p_{k})\in \left[u_{i}-1,u_{i}\right]$
so
$\unicode[STIX]{x1D700}_{i}$
can be roughly interpreted as percent error. For the example in Figure 8,
$\unicode[STIX]{x1D700}_{i}=0.125$
which is a relatively high value indicating potential errors in interpreting results, especially in regions with high estimates of one partner’s probability of collaboration but low estimates for the other.

## Appendix C. Application case data

Data for the application case was generated using the publicly available distribution of Orbital Federates Simulation – Python (OFSPY) (Grogan 2019). This software simulation computes cash flows obtained from an initial space systems design in a version of the multi-player game *Orbital Federates*. The software program contains a command line interface (CLI) to run specific design scenarios. Automated operational policies based on mixed integer linear programs determine how to use available space systems to observe, store, transmit, and downlink data to complete contracts and earn revenue each turn.

The spatial context is reduced to two dimensions with six sectors (1–6) and layers representing the surface (SUR), low Earth orbit (LEO), and medium Earth orbit (MEO) shown in Figure 9. Satellites move clockwise between orbital sectors each turn while ground stations remain fixed at the surface. Space-to-ground links (SGLs) require a satellite to be in the same sector as a ground station for data transfer. Inter-satellite links (ISLs) require satellites to be in adjacent sectors for data transfer. Proprietary links only permit data transfer within a player’s assets while open links permit data transfer between players as paid services.

Figure 9. The Orbital Federates context includes six sectors with surface (SUR), low Earth orbit (LEO) and medium Earth orbit (MEO) layers. Satellite and ground station elements transfer data using space-to-ground (SGL) and inter-satellite (ISL) links.

Designs evaluated under the independent strategy follow the CLI template: ofs.py -d 24 -p 3 -i 0 -s <SEED> -o d6,a,1 -f n <DESIGN> where -d 24 indicates a game with 24 turns, -p 3 indicates three players, -i 0 indicates no initial cash constraints, -s <SEED> indicates the random number generator seed (integer), -o d6,a,1 indicates to use a dynamic operations policy with a six turn horizon using an automatically computed opportunity cost for storage and a nominal penalty of 1 for ISLs, -f n indicates no federation operations policy, and <DESIGN> is the design specification.

The baseline scenario considers the design specification:

(i)
2.SmallSat@MEO6,SAR,pSGL 2.GroundSta@SUR1,pSGL

(ii)
3.SmallSat@MEO4,VIS,pSGL 3.GroundSta@SUR5,pSGL

Player 1 has no elements. Player 2 has a small satellite initially in MEO sector 6 with a synthetic aperture radar (SAR) and proprietary SGL (pSGL) and a ground station at surface sector 1 with a pSGL. Player 3 has a small satellite initially in MEO sector 4 with a visual light sensor (VIS) and pSGL and a ground station at surface sector 5 with a pSGL.

Designs evaluated under the collective strategy follow the CLI template: ofs.py -d 24 -p 3 -i 0 -s <SEED> -o d6,a,1 -f x100,100,6,a,1 <DESIGN> where -f x100,100,6,a,1 indicates to use an opportunistic federation operations policy with fixed prices of 100 for SGL and ISL, a six turn horizon, an automatically computed opportunity cost for storage, and a nominal penalty of 1 for ISLs.

Scenario A considers the design specification:

(i)
1.SmallSat@MEO5,oISL,oSGL 1.GroundSta@SUR3,oSGL

(ii)
2.MediumSat@MEO6,SAR,oISL,pSGL,oSGL 2.GroundSta@SUR1,pSGL

(iii)
3.MediumSat@MEO4,VIS,oISL,pSGL,oSGL 3.GroundSta@SUR5,pSGL

Player 1 has a small satellite in MEO sector 5 with an open ISL (oISL) and an open SGL (oSGL) and a ground station at surface sector 3 with oSGL. Player 2 has a medium satellite in MEO sector 6 with SAR, oISL, pSGL, and oSGL and a ground station at surface sector 1 with pSGL. Player 3 has a medium satellite in MEO sector 4 with VIS, oISL, pSGL, and OSGL and a ground station at surface sector 5 with pSGL.

Scenario B considers the design specification:

(i)
1.SmallSat@MEO5,SAR,oSGL 1.GroundSta@SUR3,oSGL

(ii)
2.MediumSat@MEO6,SAR,pSGL,oSGL 2.GroundSta@SUR1,pSGL

(iii)
3.MediumSat@MEO4,VIS,pSGL,oSGL 3.GroundSta@SUR5,pSGL

Player 1 has a small satellite in MEO sector 5 with SAR and oSGL and a ground station at surface sector 3 with oSGL. Players 2 and 3 are identical to Scenario A except removing the oISL modules.

Note that scenario A relies on close proximity between players to enable ISLs. The above design strings were modified in cases with only partial participation in the federation: MEO5 is replaced by MEO3 if only players 1 and 3 join a federation and MEO4 is replaced by MEO5 if only players 2 and 3 join a federation. Outputs reported in Table 7 compute net present values using a discount rate of 2% (per turn) averaged over the first 1000 seeds (from 0 to 999).

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