f(i→j) ⇄ f(j→i)
Bargaining power is a tug between two psychological forces.
Virtual Bargaining with Asymmetric Bargaining Power (VB-ANBS) is a theoretical chapter from my PhD thesis with a simple core picture: whenever two agents interact under some bargaining procedure — like exchanging messages before deciding — each exerts an inducing force on the other, a push to move them, while the other responds with a resisting force, a counter-push against being moved.
Put it succinctly, your bargaining power is your share of the total push.
Scroll — the diagram builds the picture piece by piece; then you can test the math yourself, play with the chapter's applications, and leave with two open questions about AI.
Two agents, i and j, meet under a bargaining procedure.
f(i→j) — i’s inducing force pushes on j.
f(j→i) pushes back. Evenly matched: Φ = 1.
Now f = 6 vs 3: Φ = 2, α_i ≈ 67% — i holds the influence.
AI enters the channel — bearing the exchange, or swallowing it?
01 · The picture
Two agents, one interaction
Call them i and j: two people negotiating a price, a leader and a follower, any pair about to coordinate on something. Between them sits a bargaining procedure — most simply, the chance to exchange messages before anyone commits to a choice.
02 · First force
The inducing force
The inducing force, written f(i→j), occurs before they make a choice. It is the "push" agent i exerts on agent j — everything i does that pulls j toward i's preferred outcome: a persuasive message, a promise, a threat.
03 · Counter-force
The resisting force
The resisting force, f(j→i), is the "counter-push": how strongly j reacts against being influenced, holding their own ground. When the two forces exactly offset, the force ratio Φ equals 1 — the push lands, but it doesn't move anyone (a stalemate).
04 · The measure
Bargaining power
Your bargaining power is your slice of the total (psychological) forces. In the chapter's notation:
αi = fi→j / (fi→j + fj→i) Φi→j = fi→j / fj→i
When Φ tips past 1, i has an influence — and a first-mover advantage. On the diagram: forces of 6 against 3 give i two-thirds of the power, and a spark marks where the forces meet.
05 · Enter AI
“The water that bears the boat is the same that swallows it up.”
The chapter reaches for this idiom when it shows how the ability to communicate can complement or nullify a force. It is not surprising to see that AI is now the water under this whole picture: the same channel that carries your push can carry it away. Whether it bears the exchange or swallows it is exactly what this page leaves open. First, get the math in your hands — then see the theory at work — and leave with two questions about AI that neither the chapter nor the evidence has settled yet.
Hands on · f(i→j) vs f(j→i)
Test the theory
Two agents, two forces, your fingers. Set each force and watch the force ratio Φ, the power split α, and the balance shift live — Φ > 1 means that side holds the influence and the first-mover advantage.
From the chapter · applications
The theory at work
The chapter takes this force picture beyond a single pair — into coordination games, threats, and groups. Each application has its own page, with the chapter's argument laid out and an interactive panel to play it on.
Application 1 · Communication & leadership
One-way or two-way talk?
Why one-way messages out-coordinate two-way conversation (Cooper et al., 1989), why symmetric talk isn't symmetric power, three ways a deadlock breaks — and what this says about authoritarian vs democratic leaders.
Play with the channel ⟶Application 2 · Strategic influence
Can the threat land?
Speaking first makes an incumbent's threat an effective deterrent; speaking first lets the threatened nullify it — and a cheap "boobytrap" can flip the logic entirely. The home of the water-and-boat idiom.
Hold the channel ⟶Application 3 · Groups & hierarchy
The fading announcement
A leader's push is finite: spread over more followers, or sent down more levels of hierarchy, each person feels less of it — the chapter's two negative associations, on sliders.
Stretch the push ⟶Where the chapter stops · AI
Two open questions
Open question 1: AI in the procedure
What happens when both sides of a negotiation let AI draft their words? If the negotiation repeated many times, would people learn to trust AI-drafted promises less — or find ways to signal sincerity despite AI's involvement?
Open question 2: AI as an agent
Across a long conversation with an AI, does your resisting force erode — or hold? Does it drift down by default, or only when you let it — and what would it take to notice the drift in the moment?