What each graph is actually showing, and how strategic voting is modeled in each system.
In a Condorcet-style pairwise system, a sincere ballot of A>B>C is really three
separate pairwise votes cast at once: A beats B, A beats C, and B beats C. States in the graph are
named by the ranked list of these matchup margins, smallest first — for example
⟨A→B ∣ B→C ∣ C→A⟩ means A's win over B is the narrowest margin, B's win over C is
larger, and C's win over A is the largest. If a Condorcet winner exists (someone who wins both of
their matchups), they win outright. Otherwise the three candidates form a cycle, and the winner is
decided by the weakest link: whoever lost their closest matchup by the smallest margin.
An insincere deviation here just means reversing one of your three pairwise votes — voting
B>A, C>A, or C>B instead of your true preference. A
coalition doing this can either weaken a matchup they'd otherwise win comfortably (concordant
with their sincere preference) or strengthen one they'd otherwise lose (discordant). It turns
out this is only ever profitable from exactly two cyclic states — the "gates" — with four further
states able to reach a gate through a longer chain of moves.
Instant-runoff nodes track two things at once: an ordering of first-place vote totals, and an
underlying pairwise "tournament" between the candidates. These don't have to agree — first-place
votes don't necessarily predict who wins head-to-head. (In Burlington's 2009 mayoral election, Kurt
Wright led in first-choice votes but lost both of his pairwise matchups.) A node looks like
⟨X<Y<Z ∣ X→Y, Y→Z, X→Z⟩: X has the fewest first-place votes and is eliminated
first, their votes transfer, and the winner is whoever wins the resulting head-to-head — Y if Y
beats Z, otherwise Z.
A sincere ballot A>B>C does two things simultaneously: it adds to A's first-place
total, and it feeds the tournament (contributing to A beating B, A beating C, and C beating A). As
with minimax, a deviation can shift either piece independently — flip a tournament matchup, or change
your declared first choice. Changing your first choice can do one of two things: drop a candidate
below whoever they were just ahead of, or, if two candidates are close in votes, swap their order.
Real elections correlate first-place strength with pairwise strength, but here we deliberately treat
them as fully independent so we can map every manipulation that could be optimal for a coalition
under the most favorable conditions — the worst case for the system.
STAR nodes share the same structure as IRV — a vote-total ordering plus a tournament, written the
same way ⟨X<Y<Z ∣ X→Y, Y→Z, X→Z⟩, with the same winner rule (Y if Y beats Z, else
Z), except here the ordering is by total score and the runoff is between the two highest scorers.
We assume a coalition sincerely prefers A>B>C and scores their ballots
5-x-0, for some star rating x strictly between 0 and 5 that may vary voter
to voter. To map the full range of possible manipulation, the coalition can switch to one of three
insincere scoring patterns:
All three preserve the tournament margins exactly, while still letting the coalition manipulate who makes the runoff. The only adjacent swap in total score these moves can't produce is moving A above C when C was just ahead of A. This makes STAR's manipulable surface strictly larger than IRV's: by flooding or starving a candidate of stars, a coalition can force almost any adjacent swap in the score ranking while leaving the tournament margins completely untouched.