Structural arbitrage on probability trees enforces that marginals must come from some joint story. This chapter makes that story explicit: correlation, joint probabilities, and why treating ten Kalshi contracts as ten independent Kelly bets is how accounts die.
Joint versus marginal
Marginal probability is one event ignoring others—P(PA wins) = 0.52 on the screen. Joint probability is a combined world—P(PA wins and Senate flips). Conditional probability P(A given B) links them: P(A and B) = P(A | B) × P(B). Markets usually print marginals per contract; portfolio risk lives in the joint you cannot see.
Independence and correlation
If A and B are independent, P(A and B) = P(A) × P(B). If P(A)=0.60 and P(B)=0.50, joint both is 0.30. Election contracts share poll shocks, turnout models, and national mood—correlation is positive more often than zero.
Positive correlation: both YES legs win or lose together. Negative correlation: one YES makes another less likely when paths conflict. Positive ρ fails diversification—losses cluster. Near-zero ρ lets independent math approximate. Negative ρ can hedge but rarely appears without explicit structure.
You rarely observe ρ directly; infer from narrative and how prices co-move on past shocks.
Joint table intuition
Suppose worlds for PA and MI give joint both-win 0.35, PA-only 0.10, MI-only 0.08, neither 0.47—rows sum to one. Marginals might be P(PA) = 0.45 and P(MI) = 0.43. If independent, both-win would be 0.45×0.43 ≈ 0.19; actual 0.35 shows positive correlation—a “blue wall” story makes long YES on both a bigger bet than two Kelly legs imply.
Conditionals and incoherence
Conditional contracts read P(general | nominee) = 0.55 with P(nominee) = 0.40, so joint path ≈ 0.22. If unconditional general YES trades 0.30 for the same story, gap may be arb, different resolution strings, or timing—check rules before size.
Portfolio variance roughly adds diagonal terms w_i² p_i(1−p_i) plus cross terms 2 w_i w_j Cov(i,j). Half-Kelly on ten correlated legs behaves like over-betting one fat leg.
Prediction-market correlation map
Same election night shares exit polls and recount risk—one risk budget. Fed path and recession timing share macro surprises. Same athlete props share injury news. Token and SEC approval share regulatory headlines. Parent nominee and general election are path-dependent—use conditionals when the tree matters.
Structural versus naive hedge
You hold YES “party wins presidency” exposure $2k and NO “party wins Senate” $1.5k. Naive view: opposite chambers hedge. Joint view: wave elections align party success—you may double wave risk if both legs are same-party outcomes. Ask whether shocks align, whether resolution clusters one night, and whether Kelly sums should be quartered into one party cap.
Fréchet bounds
Given only P(A) and P(B), joint P(A and B) must lie between max(0, P(A)+P(B)−1) and min(P(A), P(B)). With P(A)=0.60 and P(B)=0.70, joint both is between 0.30 and 0.60. If your story needs 0.45 but the cap is 0.40, beliefs are incoherent—fix before trading.
Time correlation
Prices today correlate with prices tomorrow on the same contract—autocorrelation breaks “independent daily Kelly.” News jumps move whole clusters; reduce adds per headline.
Practical habits: tag each position with a narrative cluster; sum notional per cluster with a hard cap (e.g. 8% bankroll); multiply marginals only with evidence of independence; prefer conditional markets when path matters; watch persistent cross-leg gaps for rule mismatch; stress-test “if PA YES loses, what else moves?”; log conditional forecasts separately in Brier and log journals.
Mistakes: diversification illusion across one story; parlay intuition without ρ; treating primary and general as independent draws; arb math across different resolution universes.
Copula intuition without formulas
You do not need a copula library to trade. You need a story: if the same shock moves five contracts, one headline moves your whole book. Historical correlation of price changes is a noisy proxy for joint risk—use it when narrative and data agree.
Conditional markets as joint honesty
When venues list “IF nominee X, THEN wins general,” the conditional quote is the right object for path-dependent risk. Unconditional general YES can trade higher or lower than the product of marginals depending on alternate paths. Joint thinking says: write the tree, then assign mass to branches—not to headlines.
Kelly on clusters
Summing per-contract Kelly on ten Senate races might suggest 40% of bankroll; cluster cap at 8% treats the election as one draw. That is not conservative for its own sake—it is coherent with positive correlation. Module 04.6 fractional Kelly and this chapter’s cluster cap work together.
Parlay intuition versus joint math
A sports parlay multiplies conditional prices if legs are independent; correlation makes the product fiction. Prediction-market “parlays” are often ten separate YES tickets on one narrative—worse than a labeled parlay because risk is hidden. Joint math says: one shock, one P&L day. Size once.
Structural arb as joint enforcement
Bots enforce that branch probabilities on a tree sum to one. Your portfolio is a private tree whether or not the exchange drew it for you. If your implicit joint story is impossible under Fréchet bounds, fix beliefs before Kelly. If the market’s tree disagrees with yours, you may have edge—or you may be reading different contracts.
Stress question before size
Ask: “If this contract loses, which other open positions lose with it?” If the answer is “most of them,” you are not diversified—you are levered to one story. Write the answer next to cluster tags in your journal. Joint thinking is that single habit, not a graduate course.
Negative correlation rarity
Truly negative correlation between two YES contracts on the same night is rare without explicit structure (mutually exclusive winners in one race). Do not assume hedge because narratives sound opposite—“Dem Senate, GOP House” can still correlate positively on national mood. Check prices during past shocks, not only words.
Joint and EV together
Portfolio EV is the sum of leg EVs only when you are not double-counting the same belief twice. If one model drives five tickets, EV adds arithmetically but risk does not divide by five. Joint thinking is the brake on summed EV enthusiasm.
One election night
Imagine open YES on president, Senate, and governor for the same party. One exit poll shock moves all three. Your joint world is “wave” versus “split”—assign mass to those stories, then check whether implied marginals from prices fit Fréchet bounds. If not, arb bots or you may trade the tree; if yes, you are intentionally levered—size once.
Summary
Marginals on screens are not portfolio truth. Conditionals and Fréchet bounds keep joint stories honest. Cluster caps turn that math into bankroll survival.
What comes next
Next: probability fallacies—the cognitive bugs that break joint math and calibration together.