Advantage is one of those concepts that is both very familiar to the average person, yet seemingly absent any attempts to quantitatively define it. We all roughly know when we are in a “more advantageous” position, but struggle to define exactly “why” our position is more advantageous (relative to the previous position). In this article, we will attempt to define mathematically what is advantage.
In its simplest form, advantage is the positive difference in utility $U$ between a currently selected option $O_A$ and some alternate option $O_B$:
Conversely, a disadvantage occurs when the currently selected option is worse than an alternative. In this case, the difference in utility is negative:
This of course assumes that the options being compared are of the same option type. What that means is they come from the same utility function (i.e. all travel options calculated from the travel utility function).
If you want to calculate advantage between different types of options — such as travel plans, investment decisions, and health strategies simultaneously — you are no longer dealing with a single utility function. Instead, you can think in terms of a utility vector that aggregates different utility dimensions:
\[\vec{V}_{u1} = \begin{bmatrix} U_{\text{travel}}(O_A) \\ U_{\text{finance}}(O_B) \\ U_{\text{health}}(O_C) \end{bmatrix}\]Here, $\vec{V}_{u1}$ represents the multi-dimensional utility vector associated with a particular decision set, where each component corresponds to a different utility function (or domain).
To compare two such options across multiple dimensions, we can compute the vector difference:
\[\Delta \vec{V} = \vec{V}_{u1} - \vec{V}_{u2}\]The resulting vector describes the direction and magnitude of advantage (or disadvantage) across each domain. You can also compute a scalar advantage by taking the dot product with a weighting vector $\vec{w}$ representing the relative importance of each domain:
\[A = \vec{w} \cdot \Delta \vec{V}\]This scalar $A$ gives you an overall weighted advantage, suitable for decision-making when utility components span different types.
This mathematical definition of advantage naturally extends to the context of games, especially competitive ones. In such settings, we can define a concept we will call position advantage: the relative utility of a player’s current game state compared to alternate possible states.
In other words, a game position has advantage if, from a strategic or probabilistic perspective, it leads to higher expected utility (e.g. winning the game, gaining points, achieving a goal) than some other position. Conversely, a disadvantageous position is one from which fewer favorable outcomes are accessible or more constraints are imposed.
This connects tightly to ideas in:
We can formalize this idea using expected utility over possible outcomes from a position, denoted $P$:
\[\text{Advantage}(P) = \mathbb{E}[U \mid P] - \mathbb{E}[U \mid P_{\text{baseline}}]\]Where:
This definition makes clear: advantage is not about winning outright, but about increasing the expected value of outcomes from your current position, relative to some reference.
Regardless of what is the utility function used to calculate the various options available (or in a game what we might call positions) advantage (or disadvantage) is merely the difference between the options / positions. The whole of decision making and game theory can ultimately be reduced down to determining which option(s) / position(s) relative to my current option / position maximize the positive utility difference.