Nobody Wants to Go First

How to Read the Symbols

The letters below are short names for the teams, their choices, and their points. You can read each equation as a sentence. The math reading guide gives more examples.

Team labels: \(i\), \(j\), and the small numbers below a letter

Both \(i\) and \(j\) stand for a team number, from 1 to 8. A small label below a letter is a subscript: \(a_3\), read “a sub three,” means Team 3’s choice. It does not mean “a times three.”

Choices: \(a_i\in\{0,1\}\) and \(z_i\)

Read \(\in\) as “is one of.” The braces list allowed values: 0 means wait and 1 means adopt. The separate label \(z_i\) is 1 if that team is a pilot and 0 if it is not. Multiplying by 0 removes a term; multiplying by 1 keeps it.

Points and shares: \(B\), \(C\), \(S\), \(q_i\), and \(u_i(a)\)

\(B\) is the full benefit, \(C\) the switching cost, and \(S\) the support per pilot adopter, all in points. \(q_i\) is the fraction of Team \(i\)’s seven peers who adopt. \(u_i(a)\), read “u sub i of a,” is that team’s resulting points. The \(a\) in parentheses means “using everyone’s current choices,” so those parentheses show the input to a rule.

Adding a group: \(\sum_{j\ne i}a_j\)

The large \(\sum\), called “sigma,” means add the listed values. The label \(j\ne i\) says “use each team j except team i”; \(\ne\) means “not equal to.” Adding the seven peers’ 0s and 1s counts the adopters. The shorter \(\sum_i\) later on means add across all eight teams.

Multiplication and comparisons

Letters next to each other mean multiply: \(Bq_i\) is \(B\times q_i\), and \(z_iS\) is \(z_i\times S\). A fraction bar means divide. The sign \(>\) means “is greater than”; equality means the two choices tie. See operations and comparisons.

There are eight teams. The three designated pilot teams have \(z_i=1\); the other teams have \(z_i=0\). Your team is not a pilot. Utility means the points a team gets. These formulas calculate actual peer adoption and utility:

Read it aloud: “Count the other adopters and divide by seven. Then multiply the full benefit by that share, subtract the cost, and add any pilot support. A team that waits gets zero.” For a nonpilot with five adopting peers, this is \(10\times(5/7)-6\), about 1.14 points.

Waiting earns zero. For \(B>0\), adoption is strictly preferable when \(q_i>(C-z_iS)/B\), with indifference at equality. With default values and no support, the smallest profitable number of adopting peers is five. At the defaults, both all-wait and all-adopt are strict Nash equilibria of the actual payoff game.

That comparison says: “If the full benefit is above zero, the peer share must be greater than the cost left after support, divided by the full benefit.” With no support, that share must exceed \(6/10\), or 60%.

Reading the Forecast and Update

Review labels: \(t\), \(t+1\), and \(q_i^t\)

\(t\) is the current review number; \(t+1\) is the next review. In this formula, the raised \(t\) in \(q_i^t\) labels the review. It is not a power: read it as “Team i’s peer share at review t.”

Estimates: \(\widehat q_i^{\,t}\) and \(\widehat v_i^{\,t}\)

The little roof, called a “hat,” means an estimate. \(\widehat q_i^{\,t}\) is the share the team expects; \(\widehat v_i^{\,t}\) is the points it expects from adopting. The control \(e\) changes the expected share: \(e/100\) turns 20 percentage points into 0.20.

Bounds: \(\min\) and \(\max\)

Minimum and maximum mean “pick the smallest” and “pick the largest” of the listed numbers. Read the inner brackets first. Here they keep the estimate between 0 and 1: a share cannot be below none or above everyone.

For review \(t+1\), every team forecasts from the same previous set of choices. The control \(e\) adds an expectation shift in percentage points:

Read it aloud: “Add the expectation change to the observed share, keeping the answer between zero and one. Use that estimated share to work out the points from adopting.” For example, 40% plus 20 percentage points gives an expected 60%; 90% plus 20 is capped at 100%.

The update adopts if \(\widehat v_i^{\,t}>0\), waits if it is negative, and retains the previous action on a tie. Updates are simultaneous. Assigned commitments force adoption in reviews 1–3. A user choice overrides Team 1’s update for that review; the reconsider button applies the same update rule to Team 1 as to the other teams.

Reading the Group Total

Capital \(V\) means a total value in points. The words below it are labels: \(V_{\text{teams}}\) is the teams’ total, and \(V_{\text{after funding}}\) includes the sponsor’s cost. Each \(\sum_i\) adds across all eight teams.

Read it aloud: “Add all the teams’ points. For the total after funding, subtract the support paid to pilot teams that actually adopt.” The product \(z_i a_i\) is 1 only when a team is both a pilot and an adopter. If two pilots adopt with 3 support points each, subtract \(3\times2=6\) points for the sponsor.

Support is a transfer. Its direct contribution cancels from value after funding, although changed actions can change that value. Neither aggregate includes administrative costs, unmodeled benefits, or distributional preferences.

This is a complete network with identical benefits and costs. A seeded assignment sets pilot, initial-adopter, and committed teams; subsequent reviews are deterministic given the settings and user choices. Reviews revise plans rather than accumulate payments. Commitments are forced plans, not modeled contracts. Forecasts can be wrong, and simultaneous updates can oscillate. A fixed point with distorted expectations need not be a Nash equilibrium of the actual payoff game.

Concept reference: Easley and Kleinberg, Networks, Crowds, and Markets, Section 6.5 on coordination games. The eight-team planning model and its support rules are defined here.