Combination markets sell a whole story in slices: nomination, battlegrounds, national outcome; rate decision, recession timing; game winner, series winner. Platforms rarely ship a diagram—you infer the tree from series pages, related-market links, and cross-references in the rule PDF. This chapter is topology: parents, children, siblings, and the inequalities that keep probabilities honest.
Tree vocabulary
A parent is the broader event—“party wins the presidency.” A child narrows the path—“Candidate A wins the nomination.” Siblings at one depth should be mutually exclusive—{A, B, C} in a winner-take-all primary. A leaf is a fully specified terminal state, sometimes absurdly specific (“wins Pennsylvania by 2+ points”).
Draw the graph before you trade the ticker. Arb lives in the graph, not in the slogan.
Monotonicity—when inequalities apply
If child and parent describe the same candidate path in the same cycle, the child’s probability should not exceed the parent’s on compatible definitions: you cannot be more likely to win the presidency than to win the nomination on that path. Earlier deadlines should not price higher than later ones on the same event definition—“cuts by June” at 70% and “cuts by December” at 62% is a red flag if December includes June outcomes under identical cut language.
Electoral college breaks naive trees: swing-state binaries do not sum to national win probability under winner-take-all rules. Use state slates for within-family arb, not amateur arithmetic on paths to 270.
Sibling sums and overround
Three nominees at 45%, 38%, and 22% sum to 105%—five points of overround. Compression toward 100% is the same force as vig on exclusivity; prediction markets with active bots tighten faster on deep books than on capped retail menus.
When sibling sums exceed 100% on executable asks, selling the set (or buying NO on each leg where shorting exists) is the classic fix—if exclusivity is truly stated in the rules.
Parent–child vs explicit conditionals
Many trees embed “given nomination” as a named child rather than a separate conditional product. Scan both the related tab and explicit “given” markets; math links multiply factors, UI links hide them.
Illustrative check: nomination 40%, conditional presidency 62%, standalone presidency 22%. Product 0.40 × 0.62 = 0.248 vs 22%—gap may be fees, timing, or trade; standalone rich versus tree cheap is the structural story.
Time monotonicity example
“Fed cuts by June” at 70% and “Fed cuts by December” at 62% violates time logic if both use the same cut definition—December includes June paths. Buying the longer window and selling the shorter can be arb if rules match; if June means 25bp only and December means any cut, there is no arb—fix the graph.
Electoral college: when naive trees break
Swing-state binaries do not sum to national probability. A model needs joint paths to 270, not the maximum of state marginals. Trees still help within exclusive state-winner families listed under the same rule family.
Maker dynamics on trees
Market makers quote coherent siblings; one mis-posted child can be sniped while inventory on parents lags. Thin children are cheap to move for narrative—watch trade size on leaves.
Cabinet micro-trees
Under “party controls the Senate,” committee-chair props may sum above the parent if chairs are not exclusive given control. The exclusivity footnote is the whole trade.
Design failures
Intern mis-posts fade quickly on deep books. Ambiguous parents invite dispute rather than size. Missing branches are model risk—you cannot arb a state that is not listed.
Drawing the graph in ten minutes
Screenshot related tickers, label parent and children, mark exclusive siblings, note dates on each node. Run sum checks on each exclusive layer. Run multiply checks on nomination × conditional general paths. Flag gaps larger than your fee budget.
Structural versus local arb
Sibling sum fixes are local. Parent-child packages may need multi-leg trades across the tree—structural arbitrage in the language of earlier modules. Deep trees on regulated books tighten quickly; newly listed crypto siblings may stay wide for hours.
Void when candidates withdraw
A child market on a withdrawn candidate may trade stale until halted. Parent markets may still be live. Do not arb dead children against live parents without reading halt rules.
Narrative trades versus graph trades
“Italian restaurant” stories about electability are not substitutes for checking P(child) ≤ P(parent) on the same path. When the graph is clean, math leads; when rules diverge, exit.
Relation to categoricals and conditionals
A categorical slate is often one layer of a tree exposed in the UI. Conditionals are edges between layers. Combination thinking connects both: siblings sum to one; edges multiply to joints.
Core concepts to remember
Draw the graph. Check sibling sums near 100%. Check child ≤ parent on the same path. Do not sum swing states into national win without a path model. Read void behavior on withdrawn names.
Speed versus tightness across venues
Deep regulated trees on major elections compress sums quickly. Newly listed crypto siblings may sit mispriced longer. The logic is identical; the half-life of obvious gaps differs.
Common mistakes
Child priced above parent on the same path. Summing swing states to national win. Ignoring withdrawn candidates on sibling sets. Importing one venue’s parent price to another’s child without a rule diff.
What comes next
You mapped nodes and edges. Bundled contracts package multiple nodes into one ticket.
Next: Bundled Contracts and Baskets