{
 "approach": "The base strategy, unchanged. Tiny 0.01-share probes plus a sub-tick grid filter locate p to about a tick. Quotes sit one tick inside the competitor, sized by a marginal rule (one more share while expected retail edge exceeds expected arbitrage loss). Orders rest for queue priority, and below 5c it buys YES to cover low asks. I scored it run by run on seeds 0-599. The worst 60 fall into two groups: (1) about 46 one-tick-spread runs, mostly settling at 1, that earn about 0 because no tick near p pays against arbitrage; (2) about 7 tail losses of 26 to 417, each one jump lifting a large ask book resting at 2-10c, often late in the run. I tried rules for both: joining the competitor level in 1-tick spreads, and a separate jump-loss weight at low prices (global, or only in the last 150 steps). Every rule that helped the worst 60 cost more in the rest. The base's discounting of low-price jump risk (kL=0.15) is close to optimal in expectation, so I froze the base code. This item's finding is that the worst runs are structural or are the priced-in cost of the near-zero edge.",
 "blocked": null,
 "python_version": "3.14.4",
 "base_mean_edge": 78.85441957511699,
 "score": {
  "mean_edge": 87.41650726068066,
  "retail_edge": 116.13125849331949,
  "arb_edge": -28.714751232636335,
  "simulations": 600,
  "seed_start": 0,
  "failures": 0
 },
 "holdout": {
  "mean_edge": 85.46780344973116,
  "simulations": 600,
  "seed_start": 20000,
  "failures": 0
 },
 "code": "\"\"\"Stale-quote defence with tick probes, a sub-tick belief and YES ammunition.\n\nEvery step the arbitrageur removes any order on the wrong side of p, so tiny\ntwo-sided probes around the estimate locate p to a tick; a grid filter narrows\nthat further.  Quotes sit one tick inside the competitor and are sized by the\nmarginal rule: one more share is worth quoting while the chance a retail order\nreaches it, times its edge, exceeds the expected arbitrage loss.  That rule pulls\nquotes by itself when a move would hurt most.  Near p=0 a retail buy pays almost\nits whole notional as edge, so YES is held to cover low asks.\n\"\"\"\nimport math\nfrom orderbook_pm_challenge.strategy import BaseStrategy\nfrom orderbook_pm_challenge.types import CancelOrder, PlaceOrder, Side\n\nP = dict(lam_r=0.25, m=5.0, sig_n=1.2, lam_j=0.0019, sig_j=0.35, sig=0.02,\n         qmax=3000.0, rmin=2, kA=0.7, kL=0.15, pL=0.05,\n         ammo=1.0, ydem=0.4, W=200.0, fy=0.6, ammo_q=60.0, ammo_pmax=0.05, ammo_maxpay=0.02,\n         fb=0.9, fu=0.5, xu=0.1, sticky=1.0, min_ahead=2.0, keep_hi=1.5, keep_lo=0.7, Hu=300.0, bins=8.0, filt=1.0, uni=0.01, jumpinfo=1.0, dmin=0.0, kLb=0.08, kLf=0.08, pinf=1.0, pre=1500.0, H1=100.0)\nSQ2 = math.sqrt(2.0)\nSQ2PI = math.sqrt(2.0 * math.pi)\ndef cdf(x): return 0.5 * (1.0 + math.erf(x / SQ2))\ndef pdf(x): return math.exp(-0.5 * x * x) / SQ2PI\ndef inv(p):\n    p = min(max(p, 1e-9), 1 - 1e-9)\n    t = math.sqrt(-2.0 * math.log(min(p, 1 - p)))\n    x = t - (2.515517 + 0.802853 * t + 0.010328 * t * t) / (1 + 1.432788 * t + 0.189269 * t * t + 0.001308 * t ** 3)\n    x = -x if p < 0.5 else x\n    for _ in range(2):\n        d = pdf(x)\n        if d < 1e-12:\n            break\n        x -= (cdf(x) - p) / d\n    return x\ndef lpm(k):\n    return pdf(k) + k * cdf(k)\ndef fl(q):\n    return math.floor(q * 100 + 1e-9) / 100\n\nNODES = [(-2.2, .06), (-1.3, .17), (-0.6, .27), (0.6, .27), (1.3, .17), (2.2, .06)]\nPROBE = 0.01\n\n\nclass Strategy(BaseStrategy):\n    def __init__(self):\n        self.n = 0\n        self.ahead = set()     # main orders placed strictly ahead of any competitor order at their tick\n        self.res = {}          # order id -> uncovered quantity at placement\n        self.bc = None         # belief: sub-tick grid centres (ticks)\n        self.bw = None         # belief: weights\n        self.st = 0.5          # last one-step sd in ticks\n        self.pcb = self.pca = None\n        self.pb = set()\n        self.pa = set()\n\n    def _oid(self, tag):\n        self.n += 1\n        return tag + str(self.n)\n\n    def on_step(self, state):\n        H = max(1, state.steps_remaining)\n        cb = state.competitor_best_bid_ticks or 0\n        ca = state.competitor_best_ask_ticks or 100\n        lo, hi = cb, ca\n        for o in state.own_orders:\n            if o.side is Side.BUY:\n                lo = max(lo, o.price_ticks)\n            else:\n                hi = min(hi, o.price_ticks)\n        if P['pinf'] > 0:\n            sb = {o.price_ticks for o in state.own_orders if o.side is Side.BUY and o.remaining_quantity <= PROBE + 1e-9}\n            sa = {o.price_ticks for o in state.own_orders if o.side is Side.SELL and o.remaining_quantity <= PROBE + 1e-9}\n            ea = self.pa - sa\n            eb = self.pb - sb\n            if sa and ea:\n                lo = max(lo, min(max(ea), hi))\n            if sb and eb:\n                hi = min(hi, max(min(eb), lo))\n        if P['jumpinfo'] > 0 and self.pcb is not None:\n            if ca > self.pca + 1:\n                lo = max(lo, ca - 1)\n            if cb < self.pcb - 1:\n                hi = min(hi, cb + 1)\n        self.pcb, self.pca = cb, ca\n        if lo > hi:\n            lo, hi = hi, lo\n        # sub-tick belief over p_(t-1): predict with one-step noise, truncate to [lo, hi]\n        wid = max(hi - lo, 1e-6)\n        nb = max(2, min(48, int(round(wid * P['bins']))))\n        cs = [lo + (i + 0.5) * wid / nb for i in range(nb)]\n        if self.bc is None or P['filt'] <= 0:\n            ws = [1.0 / nb] * nb\n        else:\n            sg = max(self.st, 0.6 * wid / nb)\n            k = -0.5 / (sg * sg)\n            ws = []\n            uni = P['uni'] * sg\n            for c in cs:\n                t = uni\n                for ci, wi in zip(self.bc, self.bw):\n                    d = c - ci\n                    d2 = d * d * k\n                    if d2 > -12.5:\n                        t += wi * math.exp(d2)\n                ws.append(t)\n            tot = sum(ws)\n            ws = [w / tot for w in ws]\n        self.bc, self.bw = cs, ws\n        mean_t = sum(c * w for c, w in zip(cs, ws))\n        var_t = max(sum(c * c * w for c, w in zip(cs, ws)) - mean_t * mean_t, 0.0)\n        phat = min(max(mean_t / 100.0, 0.002), 0.998)\n        var = P['sig'] ** 2 + P['lam_j'] * P['sig_j'] ** 2\n        sd = math.sqrt(H * var)\n        z = inv(phat)\n        Z = z * sd\n        sp = pdf(z) / sd * P['sig']\n        self.st = sp * 100.0\n        s = math.sqrt(sp * sp + var_t / 1e4 + 1e-8)\n        sd1 = math.sqrt(max(1.0, H - 1.0) * var)\n        jp = [(cdf((Z + n * P['sig_j']) / sd1), w) for n, w in NODES]\n        lj = P['lam_j']\n\n        def AB(x):\n            k = (x - phat) / s\n            a = s * lpm(k)\n            b = a - (x - phat)\n            ja = sum(w * max(0.0, x - p) for p, w in jp)\n            jb = sum(w * max(0.0, p - x) for p, w in jp)\n            return (1 - lj) * a + lj * ja, (1 - lj) * b + lj * jb\n\n        if H == 1:\n            def AB(x):\n                return (1 - phat) * x, phat * (1 - x)\n        rho = P['lam_r'] * 0.5\n        mu = math.log(P['m']) - 0.5 * P['sig_n'] ** 2\n        kA = P['kL'] if phat < P['pL'] else P['kA']\n\n        def qsize(loss, gain, px):\n            loss *= kA\n            if gain <= 0:\n                return 0.0\n            tail = loss / (rho * gain)\n            if tail >= 1.0:\n                return 0.0\n            return min(P['qmax'], math.exp(mu + P['sig_n'] * inv(1 - max(tail, 1e-6))) / px)\n\n        ptk = phat * 100\n        def qbid(t):\n            if not 1 <= t <= 99:\n                return 0.0\n            a, b = AB(t / 100.0)\n            if phat * 100 - t < P['dmin']:\n                return 0.0\n            if phat < P['pL']:\n                ytg = P['ydem'] * min(H, P['W']) / max(ca - 1, ptk + 1) * 100.0\n                a *= (P['kLb'] if state.yes_inventory < ytg else P['kLf']) / P['kL']\n            return qsize(a, b, max(phat, 0.05))\n\n        def qask(t):\n            if not 1 <= t <= 99:\n                return 0.0\n            a, b = AB(t / 100.0)\n            if t - phat * 100 < P['dmin']:\n                return 0.0\n            return qsize(b, a, t / 100.0)\n\n        bids = [o for o in state.own_orders if o.side is Side.BUY and o.remaining_quantity > PROBE + 1e-9]\n        asks = [o for o in state.own_orders if o.side is Side.SELL and o.remaining_quantity > PROBE + 1e-9]\n        sticky = P['sticky'] > 0\n        def top(orders, comp, step):\n            if sticky and sum(o.remaining_quantity for o in orders if o.price_ticks == comp and o.order_id in self.ahead) >= P['min_ahead']:\n                return comp\n            return comp + step\n        bid_t = top(bids, cb, 1)\n        ask_t = top(asks, ca, -1)\n        qb = qbid(bid_t)\n        qa = qask(ask_t)\n\n        Y = state.yes_inventory\n        if ask_t == 1 and H > P['H1'] and P['pre'] > 0:\n            qa = 0.0          # let the competitor's 2-tick ask erode instead of fronting it for nothing\n        # ammunition: hold YES so that low asks are covered\n        ptick = phat * 100\n        amt = 0\n        qam = 0.0\n        if P['ammo'] > 0 and ask_t >= 2 and phat < P['ammo_pmax']:\n            wealth = state.cash + Y * phat\n            x = max(ask_t, ptick + 1) / 100.0\n            target = min(P['ydem'] * min(H, P['W']) / x, P['fy'] * wealth / max(phat, 0.005))\n            if Y < target:\n                t = max(1, min(99, int(math.ceil(ptick + 1e-9))))\n                if bid_t <= t < ask_t and t / 100.0 - phat < P['ammo_maxpay']:\n                    amt = t\n                    qam = min(P['ammo_q'], target - Y)\n\n        if P['pre'] > 0 and ca == 2 and phat < 0.01 and H > P['H1'] and Y < P['pre'] and not qam:\n            amt, qam = 1, min(P['ammo_q'], P['pre'] - Y)\n        # order management\n        actions = []\n        free = state.free_cash - 2.0\n        keep_ask = 0.0\n        new_b = []\n        new_a = []\n        def cancel(o):\n            nonlocal free\n            actions.append(CancelOrder(o.order_id))\n            x = o.price_ticks / 100.0\n            if o.side is Side.BUY:\n                free += x * o.remaining_quantity\n            else:\n                free += (1 - x) * min(self.res.get(o.order_id, 0.0), o.remaining_quantity)\n            self.ahead.discard(o.order_id)\n            self.res.pop(o.order_id, None)\n        for o in state.own_orders:\n            if o.remaining_quantity <= PROBE + 1e-9:\n                cancel(o)\n        kept_ids = set()\n        def manage(orders, t, q, comp):\n            kept = 0.0\n            for o in orders:\n                if o.price_ticks == t and q > 0 and kept + o.remaining_quantity <= max(q * P['keep_hi'], 1.0):\n                    kept += o.remaining_quantity\n                    kept_ids.add(o.order_id)\n                else:\n                    cancel(o)\n            return kept\n        kb = manage(bids, bid_t, qb, cb)\n        ka = manage(asks, ask_t, qa, ca)\n        add_b = (qb - kb) if (kb < qb * P['keep_lo'] and bid_t != cb) else 0.0\n        add_a = (qa - ka) if (ka < qa * P['keep_lo'] and ask_t != ca) else 0.0\n\n        budget = P['fb'] * max(free, 0.0)\n        cost = add_b * bid_t / 100.0 + qam * amt / 100.0\n        if cost > budget:\n            f = budget / cost\n            add_b *= f\n            qam *= f\n            cost = budget\n        out = []\n        if amt and qam >= 0.01:\n            q = fl(qam)\n            out.append(PlaceOrder(side=Side.BUY, price_ticks=amt, quantity=q))\n            free -= q * amt / 100.0 + 1e-6\n        if bid_t >= 1 and add_b >= 0.01:\n            q = fl(add_b)\n            oid = self._oid('b')\n            out.append(PlaceOrder(side=Side.BUY, price_ticks=bid_t, quantity=q, client_order_id=oid))\n            free -= q * bid_t / 100.0 + 1e-6\n            if bid_t > cb:\n                self.ahead.add(oid)\n        if ask_t <= 99 and add_a >= 0.01:\n            x = ask_t / 100.0\n            covk = sum(o.remaining_quantity - min(self.res.get(o.order_id, 0.0), o.remaining_quantity)\n                       for o in asks if o.order_id in kept_ids)\n            cov = max(0.0, Y - covk - 0.1)\n            unc = max(0.0, add_a - cov)\n            if unc > 0:\n                ub = max(0.0, free - (1 - P['fu']) * max(state.cash, 0.0)) if x >= P['xu'] else 0.0\n                if H < P['Hu'] and phat < 0.05:\n                    ub = max(0.0, free * (1 - H / P['Hu']))\n                unc = min(unc, ub / (1 - x))\n            q = fl(min(add_a, cov + unc))\n            if q >= 0.01:\n                oid = self._oid('a')\n                out.append(PlaceOrder(side=Side.SELL, price_ticks=ask_t, quantity=q, client_order_id=oid))\n                r = max(0.0, q - cov)\n                self.res[oid] = r\n                free -= r * (1 - x) + 1e-6\n                if ask_t < ca:\n                    self.ahead.add(oid)\n\n        # probes: tiny orders that the arbitrageur removes when p crosses them\n        f = int(math.floor(ptick + 1e-9))\n        r = max(int(P['rmin']), int(math.ceil(2.0 * s * 100)))\n        self.pb = set()\n        self.pa = set()\n        if free > 1.0:\n            for t in range(f, f + r + 1):\n                if cb < t < ca and 1 <= t <= 99:\n                    out.append(PlaceOrder(side=Side.BUY, price_ticks=t, quantity=PROBE))\n                    self.pb.add(t)\n            for t in range(f + 1 - r, f + 2):\n                if cb < t < ca and 1 <= t <= 99:\n                    out.append(PlaceOrder(side=Side.SELL, price_ticks=t, quantity=PROBE))\n                    self.pa.add(t)\n        return actions + out\n",
 "code_sha256": "432d867b1459fdcae2aba14e8634d3aa8d181f3f81d16f8a8deed5252aab75c4",
 "tried": [
  "Reproduced the base: seeds 0-199 78.854420 (wave 1 78.854420); seeds 0-599 87.4165, retail 116.1313, arb -28.7148, 0 failures.",
  "Worst 60 of seeds 0-599 under the base average -16.79 (rest 99.0). 46 of them have a 1-tick competitor spread ({3: 6, 2: 6, 1: 46, 4: 2}); 48 settle at 1. Mean edge by spread/settlement: s1/0 34.4, s1/1 0.7, s2/0 133.1, s2/1 11.6, s3/0 207.6, s3/1 22.4, s4/0 289.8, s4/1 49.2.",
  "About 40 of the worst 60 are 1-tick-spread runs that settle at 1 and earn about 0 (-1.2 to +0.3). Traced seed 45: the visible spread is often 2 because of replenishment, but the free tick is within about 0.4 tick of p, so the marginal rule quotes 0 there: about 1 tick of diffusion per step against a retail rate of about 0.125 per side. There is little edge left to fix in these runs.",
  "The 7 big losers (seeds 527 -417.0, 332 -141.9, 469 -104.8, 206 -96.0, 440 -76.2, 340 -47.7, 1 -26.4) each lose in one or two steps, when a jump lifts a large ask book resting at about 2-10c. Examples: seed 527 lost 169 at step 1513 (p 0.03->0.22) and 303 at step 1992 (p->0.99 with 8 steps left); seed 332 lost 182 at step 1471 (p 0.03->0.20); seed 206 lost 207 at step 1962.",
  "Join the competitor's level in 1-tick spreads (bid at cb / ask at ca, retail gain discounted by 0.5): no change at all, 87.417 on 0-599, because the marginal rule sizes it to 0.",
  "Separate jump-loss weight kJ below 5c, replacing the kL=0.15 discount for the jump term, everywhere: kJ=1.0 gives 80.434 (worst 60 -3.36 vs -16.79); kJ=0.4 gives 85.103 (worst 60 -11.06); kJ=0.25 gives 86.454 (worst 60 -13.98). It helps the worst runs but costs more in the settle-0 runs.",
  "Jump weight kJ=1.0 only in the last 150 steps: 83.454 (worst 60 -11.64). The late low-price steps of settle-0 runs earn far more than late jumps cost.",
  "Opposite direction, jump weight halved below 5c (kJ=0.075): 87.567 (worst 60 -21.31). +0.15 is within noise (standard error about 5 on 600 runs) and it worsens the worst runs, so not adopted.",
  "--workers 4 is refused on this machine (PermissionError opening a socket), so runs went in parallel seed ranges. --sandbox works on Python 3.14.4: the base passed 20/20 (mean 90.489909); the starter also ran.",
  "Final: the base code unchanged, since no rule for the worst runs beat it beyond noise. Seeds 0-599 87.417, holdout 20000-20599 85.4678."
 ],
 "next": "Do not spend more on the worst runs. The 1-tick-spread/settle-1 runs have about 0 edge on offer, and insuring against low-price jumps costs more than it saves (kJ 0.25 to 1.0: -1 to -7). Score lives in settle-0 runs near zero. Try: (a) widening the s1/0 gap (34 vs 133 for s2/0) with the near-zero erosion and pre-buy logic for 1-tick spreads; (b) a weaker low-price jump weight (kJ 0.075 gave +0.15, noise, so test on 1200+ seeds); (c) per-run jump-rate estimation, so a high-jump run cuts low-price ask size and a low-jump run does not."
}