Table 3 Notation for Karma game and modelling
From: Karma economies for sustainable urban mobility – a fair approach to public good value pricing
Symbol | Description |
|---|---|
Indexes & scalars | |
i | Index of agent in population |
j | Index of participant in interaction |
t | Index of time/epoch of the game |
e | Index of interaction |
n | Number of agents in population |
Sets | |
\({\mathcal{N}}\) | Set of agents in population |
\({\mathcal{T}}\) | Set of possible agent types |
\({\mathcal{U}}\) | Set of possible urgency levels |
\({\mathcal{K}}\) | Set of possible Karma balances |
\({\mathcal{J}}\) | Set of participants in interaction |
\({{\mathcal{A}}}_{k}\) | Set of possible actions for participant (with Karma balance k) |
\({\mathcal{O}}\) | Set of possible outcomes from an interaction |
\({{\mathcal{B}}}_{e}\) | Vector of participants’ actions (in interaction e) |
Agent state | |
τi | Type |
\({u}_{i}^{t}\) | Urgency level |
\({k}_{i}^{t}\) | Karma balance |
Interaction | |
\({a}_{j}^{e}\) | Action of participant j in encounter e |
oe | Outcome of interaction |
\({o}_{j}^{e}\) | Outcome of interaction e for participant j |
Modelling (probabilistic functions) | |
\({\Theta }_{p}[o,{\mathcal{B}}]\) | Probability of outcome o given the participant actions \({\mathcal{B}}\) |
\({\Omega }_{p}[{k}_{j}^{t+1},{k}_{j}^{t},{{\mathcal{B}}}_{e},{o}_{j}^{e}]\) | Probability of next Karma kt+1 given current Karma kt, participant’s actions \({{\mathcal{B}}}_{e}\) and the participant’s outcome \({o}_{j}^{e}\) |
\({\Psi }_{p}[\tau ,{u}_{j}^{t+1},{u}_{j}^{t},{o}_{j}^{e}]\) | Probability of next urgency \({u}_{j}^{t+1}\) given current urgency \({u}_{j}^{t}\), outcome \({o}_{j}^{e}\), type τ |
Modelling (logic functions) | |
C[u, o] | The immediate costs for a given urgency level and outcome |
T[τ] | The discount factor for a given agent type (of temporal preference) |
Z | Karma overflow account |
\(\delta {k}_{i}^{t}\) | Karma payment (positive means receiving) |
Social State | |
πp[τ, u, k, a] | Probability of action a given the state τ, u, k |
dp[τ, u, k] | Share of population that has specific type τ, urgency level u and Karma balance k |
Optimization (intermediate products) | |
νp[a] | Probability of action a (average agent) |
γp[o, a] | Pr(o∣a) (average agent) |
κp[k*, k, a] | Pr(k*∣k, a) |
ξ [u, a] | Immediate expected cost for known action |
ρp[τ, u*, k*, u, k, a] | Pr (u*, k*∣k, u, a, τ) |
R[τ, u, k] | Expected immediate cost |
Pp[τ, u*, k*, u, k] | Pr(u*, k*∣k, u, τ) |
V[τ, u, k] | Expected infinite horizon cost |
Q[τ, u, k, a] | Single-stage deviation reward |
\({\widetilde{\pi }}_{p}[\tau ,u,k,a]\) | Perturbed best response policy |
Optimization (hyper parameter) | |
η | Change speed of πp relative to d |
ϖ | Change speed of πp |
λ | Greediness when calculating Q |
