The cooperative game as a coalition value function¶

After the initial intuition about cooperation, we now need to take an important step: describe the situation precisely.

The central idea of cooperative games is surprisingly simple.

We do not model strategies. We do not model step-by-step decisions. We do not model sequences of moves.

We model only this:

how much value each possible group of players can generate when it cooperates.

Players and coalitions¶

Consider a finite set of players

Any subset \(S \\subseteq N\) is called a coalition.

This includes:

• players alone, like \(\\{i\\}\),
• intermediate groups, like \(\\{1,3,4\\}\),
• and the grand coalition \(N\), containing everyone.

The central question becomes:

If exactly the players in \(S\) cooperate with each other, how much can they generate?

The characteristic function¶

That question is answered by a function

where \(2^N\) is the set of all possible coalitions.

For each \(S \\subseteq N\), the number \(v(S)\) represents the total value the players in \(S\) can generate by cooperating.

We can now write this precisely.

A (transferable-utility) cooperative game is described by a pair \((N,v)\), where

This function is called the characteristic function (or coalition function) of the game.

What does this function really mean?¶

Depending on the context, \(v(S)\) may represent:

• profit generated by a consortium of companies,
• cost savings from sharing infrastructure,
• votes obtained by a political coalition,
• joint production capacity,
• risk reduction from acting as a group.

The essential point is that \(v(S)\) measures the total value available to that group.

Think of \(v(S)\) as the size of the pie available to coalition \(S\). How that pie is divided comes later.

A concrete example¶

Consider three companies interested in building a gas pipeline:

Alone, none of them can make the project viable:

But in pairs, they can already extract some value:

And all three together achieve an even better outcome:

That table is the entire game. Nothing else is needed.

What we are not modeling¶

Notice what we left out:

• who proposed the alliance,
• who negotiated with whom,
• what the order of decisions was,
• what strategies were used.

All of that disappears.

What remains is only the value structure of cooperation.

Why is this powerful?¶

Because from this function \(v\) alone, we can start asking deep questions:

• How should we split \(v(N)\) among the players?
• Who is essential for generating value?
• Do stable divisions exist?
• What is a fair division?
• Who has more power within the coalition structure?

All classic concepts in cooperative game theory arise exclusively from this function.

An important change in perspective¶

In non-cooperative games, the focus is on players' behavior.

Here, the focus is on the mathematical structure of cooperation.

The game is not a sequence of actions.

The game is a table of coalition values.

And it is from that table that the whole theory is built.