For any reputation system, it is important to identify two parties:

1. Individuals, $I \in \mathcal{I}$, who earn and hold reputation within the system.
2. Communities, $C \in \mathcal{C}$, which use reputation to determine how much trust to assign to individual community members.

We assume that each community, $C$, has a notion of trustworthiness $\mathcal{T}_C$, where $\mathcal{T}_C$ is a partially ordered set with ordering $\leq$. $\mathcal{T}_C$ represents the different possible levels of trust within the community.

For a given community, $C$, a reputation system is a function

$$ R: \mathcal{I} \times \mathcal{T}_C \rightarrow \{0, 1\} $$

which satisfies the condition that, for an individual $I \in \mathcal{I}$, and for $t_1, t_2 \in \mathcal{T}_C \textrm{ with } t_1 \leq t_2$:

$$ R(I, t_2) = 1 \implies R(I, t_1) = 1. $$

For an individual, $I$, and a level of trust $t \in \mathcal{T}_C$: $R(I, t) = 1$ if $I$ can be trusted to level $t$ and is $0$ otherwise.

Example: Military Ranks

$C$ is the military in question. $I$ is an individual soldier in that military. $\mathcal{T}_C$ is the set of non-negative integers $\{0, 1, 2, \ldots\}$.

$R(I, n) = 1$ if the individual $I$ can be trusted to lead $n$ other individuals and is $0$ otherwise.

Note: This is an oversimplification of military ranks and doesn’t incorporate important information about things like medals and badges.

Example: Credit rating

$C$ is a bank, $I$ is a person applying for a loan, $\mathcal{T_C}$ is the set of possible loan terms:

$$ \mathcal{T}_C := \{L(P, r, T, c)\}, $$

where $P$ is the principal of the loan, $r$ is the annual interest rate on the loan, $T$ is the term of the loan , and $c$ is the amount of collateral on the loan. $\mathcal{T}_C$ has a partial order induced by:

1. If $P_1 \leq P_2$, $L(P_1, r, T, c) \leq L(P_2, r, T, c)$ for any $r, T, c$
2. If $r_1 \leq r_2$, $L(P, r_1, T, c) \leq L(P, r_2, T, c)$ for any $P, T, c$