Who Really Decides? The Surprising Logic of Power
A Council Vote That Could Fall Apart
The school council is stuck. Three big grades (Grade 10, 11, and 12) each have 4 votes. Two middle-sized grades (Grade 8 and 9) each have 2 votes. Grade 7 has 1 vote. To pass a budget for the spring dance, you need at least 12 votes.
Anna from Grade 12 wants to spend $1000 on a DJ. Bella from Grade 11 wants $800. Caden from Grade 10 wants $600. The medium grades want only $400. The youngest rep, Ella, thinks $500 is perfect. Everyone votes once, without knowing how the others will vote. If no budget gets 12 votes, the dance gets $0 — nobody’s idea of a good time.
This is not just a messy argument. It is a game — not a game like tag, but a mathematical model of a decision. When philosophers and logicians build such models, they uncover the hidden logic of who really has power, and which outcomes can survive.
What You Want: The Shape of Preferences
Before we talk about power, we need to know what each player wants. In logic, we describe this with a preference relation. For any two possible outcomes, a preference relation tells you whether a player likes one at least as much as the other, or whether they are indifferent.
Using the symbol ⪰ (read “is at least as good as”), we can write:
- $1000 ⪰Anna $500 means Anna thinks $1000 is at least as good as $500.
- If both $1000 ⪰Anna $500 and $500 ⪰Anna $1000, Anna is indifferent between them.
Most logicians assume three properties for a sensible preference relation. It should be reflexive (every outcome is at least as good as itself). It should be transitive (if you like A at least as much as B, and B at least as much as C, then you like A at least as much as C). And it should be connected (for any pair of outcomes, you can always compare them; you never say “I just can’t decide”).
When a preference relation is reflexive, transitive, and connected, it becomes a complete ranking. And then we can often assign a utility function — a number for each outcome, like a score — that captures the ranking. Anna might give $1000 a utility of 10, $800 a 7, and $0 a 0. You don’t need to know the exact numbers; the ordering is what matters.
What You Can Do: Power in Groups
Now we turn to the second ingredient: effectivity function. That is a fancy name for a simple idea. For any group of players — a coalition — the effectivity function lists the sets of outcomes that group can force to be the final result, no matter what the others do.
In the council game, what can Anna alone force? She has 4 votes, but that is nowhere near 12. So her effectivity is just the set of all outcomes. She can’t rule anything out. The same is true for any single player, and even for the two medium grades together (their 4 votes aren’t enough).
But the coalition {Anna, Bella, Caden} has 12 votes. They can force any single outcome they all agree on — say $1000 — by voting together. Their effectivity function includes the set {$1000} and all supersets of it. That means they can make sure the final outcome is inside any set that contains $1000.
Effectivity functions always obey outcome monotonicity: if a coalition can force a small set of outcomes, it can automatically force any bigger set that contains it. And a well-designed voting system gives an effectivity function that is truly playable. That means it satisfies several logical properties (such as never having the empty set in a coalition’s power, and certain consistency rules). Remarkably, a truly playable effectivity function is exactly the kind of power structure you get from a real game where each player chooses a strategy and the combination of their choices determines the outcome — what game theorists call a non-cooperative game.
Stable Outcomes: The Core
With preferences and powers in hand, we can ask: which outcomes will hold? One answer comes from the core — the set of outcomes that are stable because no coalition both can and wants to change them.
An outcome is in the core if there is no group of players who can force something different and who all strictly prefer that alternative. In our council, suppose the current outcome is $0 because no budget got 12 votes. Could any coalition improve? Yes: {Anna, Bella, Caden} can force $1000, and all three prefer $1000 to $0. So $0 is not stable.
What about $1000? Anna loves it, but Bella prefers $800, and Caden $600. If Bella and Caden join with the two medium grades, they have 8+4=12 votes and can force $600 (or $400). All of them prefer $600 to $1000, so $1000 is not in the core either.
In fact, some games have an empty core — no outcome is safe. The question of whether a stable outcome exists turns into a precise logical statement, something like: it is not the case that some coalition can grab a world where all its members get a strictly better deal. Logicians can express this with symbols like ◇≻ (meaning “there is an outcome that is strictly preferred”).
If Everyone Acts Alone: Nash Equilibrium
The core assumes players can openly form coalitions and make binding agreements. But many real situations are more like a game where everyone chooses an action simultaneously and cannot rely on group contracts. In that setting, the famous solution concept is the Nash equilibrium, named after the mathematician John Nash (1928–2015).
Imagine each council representative writes a budget number on a slip of paper, and a computer adds the votes. If any number reaches 12 votes, that’s the result; otherwise the dance gets $0. A strategy profile (a choice for each player) is a Nash equilibrium if no single player, by changing only her own slip, can get an outcome she strictly prefers.
Consider this profile: Anna votes $800, Bella votes $800, Caden votes $600, the medium grades vote $400, and Ella votes $500. No budget gets 12, so the outcome is $0. Is it an equilibrium? Anna could switch to $1000, but that still won’t reach 12, so the outcome stays $0 — no improvement. However, if Caden switches to $800, then $800 gets 8+4+2? Wait, Anna and Bella for $800 (8 votes), Caden for $800 adds 4, total 12. So $800 would pass, and Caden might prefer $800 to $0. In that case the original profile was not an equilibrium.
The logician Marc Pauly (working in the early 2000s) showed that you cannot fully express Nash equilibrium using the simple language of Coalition Logic alone. You need to talk about what each player can do while holding the others’ moves fixed — a kind of best response reasoning. Newer logical tools combine choice and preference into a single “intersection” modality, allowing us to say: there is no achievable world that this player strictly prefers. That’s the core of a Nash equilibrium.
Why This Matters: From Lunch Tables to Real Treaties
You might never vote on a $1000 DJ, but you face these same structures all the time. When your table argues about where to go for lunch, when your class picks a project topic, or when world leaders hammer out a climate treaty, the same logic hums underneath: what people want, what groups can achieve, and which outcomes will hold.
Knowing this can make you a sharper observer. You can spot why a decision feels “stuck” — maybe no stable outcome exists. You can see a coalition forming before anyone names it. And you learn that fairness often depends not just on people’s kindness, but on the invisible architecture of rules and votes.
Scientists and philosophers continue to build richer languages — adding knowledge, resources, probability, and time — to model even knottier decisions. But the starting point is always the same: a game, a set of minds with desires, and the question of who has the power.
Think about it
- If everyone in a group has one vote but some people are much more persuasive, does that change the logic of power? Why or why not?
- Can you think of a time when a decision was stable even though most people didn’t like it? What made it stable?
- If you could design brand-new voting rules for your class, how would you make sure the outcome is fair for everyone — and still possible to reach?
