How to use this guide
What you are looking at. A miniature course guide. A real one covers a whole business course; this one covers three chess ideas that happen to be business ideas in a costume: a decision tree and expected value, a logistic formula with a little statistics, and opportunity cost for a scarce resource. You do not need to play chess. If you can add and multiply, you can follow it.
Inside a topic: read the idea, look at the formula, then open each worked example and follow every step. Close it and redo it on paper. Then change the numbers in the lab until the answer stops surprising you, and finish with the traps.
Then practise. The Drill makes up new numbers every time. The multiple-choice bank explains why each wrong answer is wrong. The written questions have a Hint button and a model answer to compare with. The flashcards flip. Everything is levelled:
| Level | What it asks | Example |
|---|---|---|
| Beginner | A definition, or one step | Add up the material on the board |
| Intermediate | A standard problem in a few steps | Find a player's expected score from two ratings |
| Exam level | A full question that strings several steps together | Update a rating after an upset, or find a gambit's break-even chance |
| Hard | Twists: working backwards, combining topics, and the traps that cost the most marks | Work out the rating gap from a 75% score |
The question generator starts on Auto: it begins at Beginner and moves up a level after three correct answers in a row. You can pick a level yourself at any time. The multiple-choice bank and flashcards are ordered from easy to hard.
Saved answers and quiz scores stay in this browser; signed in, they also follow your account to your other device. Everything on this page is original and free to explore.
Gambits: expected value and the decision tree
A gambit is an opening where you give up material, usually a pawn, hoping the position you get is worth more than the pawn. In business terms: pay a known cost now for an uncertain payoff later. The tool for judging that is a decision tree with an expected value at the end.
Keeping score
- A game is worth 1 for a win, ½ for a draw and 0 for a loss. The average of those over many games is your expected score.
- Material uses the usual convention: pawn 1, knight 3, bishop 3, rook 5, queen 9. It is a convention, not a law of physics.
- So "the quiet line is worth 0.52" means: over many games from that position, you would collect about 52% of the points.
The decision tree
- A decision node is your choice (quiet line or gambit). A chance node is something you do not control (does the opponent go wrong?).
- At the end of each branch, write the payoff. Multiply each payoff by its probability and add: that is the expected value of the branch.
- Pick the decision with the higher expected value, unless something about the situation changes what the payoff means (see the Hard example).
Worked example 1: is the gambit worth it? original Beginner
The quiet line is worth 0.52. A gambit works 40% of the time (the opponent goes wrong and you score 0.80) and fails 60% of the time (you are a pawn down and score 0.40).
- Gambit branch: EV = 0.40 × 0.80 + 0.60 × 0.40 = 0.32 + 0.24 = 0.56.
- Compare with the quiet line: 0.56 − 0.52 = +0.04 in favour of the gambit.
- Over 100 such games, that is about 4 extra points. Not huge, but free.
Worked example 2: how often must it work? original Intermediate
Same payoffs: Q = 0.52, G = 0.80, B = 0.40. At what chance of success are you indifferent?
- Set the gambit's EV equal to Q: p × 0.80 + (1 − p) × 0.40 = 0.52.
- Rearrange: 0.40 + 0.40p = 0.52, so p = 0.12 ÷ 0.40 = 0.30 (the formula gives the same: (0.52 − 0.40) ÷ (0.80 − 0.40)).
- Check: 0.30 × 0.80 + 0.70 × 0.40 = 0.24 + 0.28 = 0.52 ✓.
- The gambit is worth playing only if you believe the opponent errs more than 30% of the time. Your estimate of 40% clears that bar; an estimate of 20% would not.
Worked example 3: a two-stage tree original Exam level
Quiet line: 0.50. Gambit: the opponent declines with probability 0.30 and the game is level (0.50). With probability 0.70 they accept; then you have a 40% chance they mishandle it (you score 0.78) and a 60% chance you are simply a pawn down (0.38).
- Work backwards from the right. The "accept" node: EV = 0.40 × 0.78 + 0.60 × 0.38 = 0.312 + 0.228 = 0.540.
- Now the first chance node: EV = 0.70 × 0.540 + 0.30 × 0.50 = 0.378 + 0.150 = 0.528.
- Compare with 0.50: the gambit is ahead by 0.028.
- Rolling back from the end of the tree to the start is the standard method for any multi-stage decision.
Worked example 4: when the payoff itself changes original Hard
In a final you must win to advance; a draw is as bad as a loss. The quiet line wins 30%, draws 50% and loses 20%. The gambit wins 40%, draws 5% and loses 55%.
- On normal scoring: quiet = 0.30 + 0.5 × 0.50 = 0.55; gambit = 0.40 + 0.5 × 0.05 = 0.425. The quiet line looks clearly better.
- But the payoff is not normal scoring. Redefine it: win = 1, anything else = 0. Now quiet = 0.30 and gambit = 0.40.
- The gambit is the right choice. Same position, same probabilities; the answer flipped because what you are maximising changed.
- In business: before computing an expected value, ask whether the thing being averaged is what you actually care about (profit, survival, hitting a deadline).
Lab: should you gambit?
History break. In London in 1851, Adolf Anderssen beat Lionel Kieseritzky in a casual game later called the "Immortal Game". It began as a gambit, and Anderssen went on to sacrifice a bishop, both rooks and his queen before checkmating with his three remaining minor pieces. It is one of the most famous chess games ever played, and a classic case of a brilliant result being mistaken for a sound decision (see the first trap).
Traps
- Outcome bias. A gambit that worked once tells you little. Judge the decision by the probabilities and payoffs at the time, not by how one game ended.
- Your probabilities are estimates. If a small change in p flips the decision, the decision is fragile; say so.
- Do not average the wrong thing: check what the payoff really is (the Hard example) before multiplying.
- The probabilities on a chance node must add to 1. Check every node.
- Expected value is a long-run average. For a single game it can be a number that never actually happens.
Ratings: a logistic curve with a job
An Elo rating turns the gap between two players into an expected score, and then nudges each rating after the game. It is a tidy example of a logistic model: a smooth S-curve from 0 to 1.
Expected score
- Your expected score against an opponent depends only on the rating gap, not on the ratings themselves.
- Equal ratings: 0.50. A bigger gap pushes the favourite towards 1 and the underdog towards 0, but never reaches either.
- It is an expected score, so draws count ½. It equals the chance of winning only if draws never happen.
- A gap of 400 points means the favourite scores about 10 points for every 1 the underdog scores.
Updating a rating
- After a game: new rating = old rating + K × (actual score − expected score).
- Beat someone you were expected to beat and you gain a little; beat someone you were expected to lose to and you gain a lot.
- K is a tuning constant. A large K reacts fast but is noisy; a small K is stable but slow. Federations use different values for new and established players.
- With the same K for both players, one player's gain is exactly the other's loss.
| Rating gap in your favour | 0 | 100 | 200 | 400 | 800 |
|---|---|---|---|---|---|
| Your expected score | 0.500 | 0.640 | 0.760 | 0.909 | 0.990 |
Worked example 1: two equals original Beginner
Two players are both rated 1500. What is each one's expected score?
- The gap is 0, and 10⁰ = 1.
- E = 1 ÷ (1 + 1) = 0.50 for each. Nothing is known to separate them.
Worked example 2: a 200-point gap original Intermediate
You are rated 1650 and your opponent 1450. What score should you expect?
- RB − RA = 1450 − 1650 = −200, so the exponent is −200 ÷ 400 = −0.5.
- 10^−0.5 = 0.3162, so E = 1 ÷ (1 + 0.3162) = 0.760.
- Over 20 games you would expect about 15.2 points, and your opponent about 4.8.
Worked example 3: an upset and the update original Exam level
A (1500) beats B (1700). K = 20 for both players. Find both new ratings.
- A's expected score: gap against A = 200, so EA = 1 ÷ (1 + 10^0.5) = 1 ÷ 4.1623 = 0.2403.
- A won, so S = 1. Change = 20 × (1 − 0.2403) = +15.19, giving A = 1515.19.
- B's expected score was 0.7597 and B scored 0: change = 20 × (0 − 0.7597) = −15.19, giving B = 1684.81.
- The two changes cancel. Had A been the favourite and won, the change would have been small.
Worked example 4: working backwards original Hard
Over a long run you collect 75% of the points against one particular opponent. About how many rating points stronger are you?
- Solve E = 1 ÷ (1 + 10^(−gap ÷ 400)) for the gap: the odds E ÷ (1 − E) = 0.75 ÷ 0.25 = 3.
- Then 10^(gap ÷ 400) = 3, so gap = 400 × log₁₀ 3 = 400 × 0.4771 = 190.8, about 191 points.
- Sanity check against the table: 200 points gives 0.760, so 191 giving 0.75 is right.
- This is the "logit" idea from regression: the log of the odds is linear in the gap.
Lab: rating gap to expected score
History break. Arpad Elo, a physics professor and strong chess player, devised the system, and the world chess federation adopted it in 1970. A famous cautionary tale about trusting a clever-looking model: in 1770 Wolfgang von Kempelen unveiled the "Mechanical Turk", a chess-playing automaton that beat many famous challengers. Inside the cabinet sat a human chess master. (Amazon later borrowed its name for a crowd-work marketplace.) Ask what is really inside any model, ratings included.
Traps
- The sign of the exponent: it is (opponent − you) ÷ 400. Get it backwards and the favourite becomes the underdog.
- Ratings are only comparable inside one pool of players. A 1500 in one federation is not a 1500 in another.
- Expected score is not win probability when draws are possible.
- A single result moves a rating by at most K points. One upset does not rewrite your rating.
- The rating is an estimate with error, like any sample statistic. A few games say little.
The clock: time as a scarce resource
Each player has a fixed bank of time for the whole game. A minute spent on one move is a minute you no longer have for another, which is the definition of opportunity cost. Good players spend time where it buys the most.
The budget
- A time control like "15 + 10" means 15 minutes at the start plus a 10-second increment added after every move you play.
- Total budget = starting time + increment × number of moves.
- Dividing by the number of moves gives an average time per move. That is a baseline, not a target.
The marginal rule
- Keep spending time on a decision while the extra expected score from one more minute is bigger than what that minute is worth later.
- Stop when the next minute buys less than it costs. The first minutes on a hard move are valuable; later ones show diminishing returns.
- Near the end of the bank the price of a minute rises, because each remaining move is played with less margin for error.
Pool model: this topic treats the whole budget as one pool shared by all your moves. A real clock adds the increment move by move, which makes the first moves a little tighter, but the pool is a good first estimate.
Worked example 1: what do you have to work with? original Beginner
A 15 + 10 game that lasts 40 of your moves.
- Starting time: 15 × 60 = 900 seconds. Increments: 10 × 40 = 400 seconds.
- Budget = 900 + 400 = 1,300 seconds (21.7 minutes).
- Average = 1,300 ÷ 40 = 32.5 seconds per move.
Worked example 2: the cost of one long think original Intermediate
Same game. You burn 6 minutes on one move. What is the average for your other 39 moves?
- 6 minutes = 360 seconds, so the pool left is 1,300 − 360 = 940 seconds.
- 940 ÷ 39 = 24.1 seconds per move, down from 32.5: that single think cost every other move about a quarter of its time.
- That drop is the opportunity cost, and it is why "I'll just think a bit longer" is never free.
Worked example 3: how long should you think? original Exam level
On a critical move, thinking lifts your expected score by an estimated amount. Each minute you keep for later is worth 0.012 points. Choose among the options:
| Think for (minutes) | Gain | Cost (0.012 × minutes) | Net |
|---|---|---|---|
| 2 | 0.050 | 0.024 | +0.026 |
| 4 | 0.075 | 0.048 | +0.027 |
| 5 | 0.080 | 0.060 | +0.020 |
| 8 | 0.090 | 0.096 | −0.006 |
- Net = gain − cost. The best net is at 4 minutes (+0.027).
- Marginal check: going from 4 to 5 minutes adds 0.005 of gain but costs 0.012, so the fifth minute loses value. Stop at 4.
- Eight minutes is worse than not thinking at all (net below zero).
Worked example 4: why the last minutes cost more original Hard
Eight moves from the end, you can arrive with 5 minutes left or with 1 minute left (you spent the 4-minute difference thinking earlier). With 5 minutes you blunder on 3% of moves; with 1 minute, on 15%. Each blunder costs about 0.25 points of expected score.
- Expected loss with 5 minutes: 8 × 0.03 × 0.25 = 0.06. With 1 minute: 8 × 0.15 × 0.25 = 0.30.
- The 4 minutes saved are worth 0.30 − 0.06 = 0.24, or 0.06 per minute.
- Compare Example 3, where a minute was worth 0.012. The same minute is worth five times as much at the end of the bank.
- So a constant "value per minute" is a simplification. Resources become more valuable as they run out, which is why scarcity prices rise.
Lab: the time budget
History break. Early tournaments used sandglasses to stop slow players, and the first mechanical double chess clock was used at London in 1883. Before that, a player with no time pressure could take as long as they liked, which is the least businesslike way to run a competition.
Traps
- An average per move is a baseline, not a rule. Easy moves should cost far less and critical ones far more.
- Do not forget the increment when you compute the budget, and do not count it twice.
- Sunk time: minutes already spent are gone. Decide each next minute on its own cost and gain.
- The price of a minute is not constant. It rises as your bank runs low.
Calculation drill
New numbers every time. Type your answer, check it, and read the solution when you miss. On Auto the generator starts at Beginner and steps up a level after three correct answers in a row. Pick a level, or a single question type, to focus.
Generator
Multiple choice bank
Original questions, ordered from Beginner to Hard. Pick an answer to see why it's right, and why the others are tempting but wrong. Use the Hint button first if you like.
Written concept questions
Some of this material is examined in words, not just numbers. Answer each one in a few sentences, tied to the situation in the question. Use Hint if you are stuck, then open the model answer to compare. Ordered from Beginner to Hard.
Flashcards
Say the answer out loud, then tap the card to check. Cards run from Beginner to Exam level; Shuffle mixes them once you know the basics.
Review sheet checklist
Writing a one-page summary is good review for any course. Every real guide ends with a checklist like this one. Tick items off as you add them to your own sheet; the ticks stay in this browser.
Side 1 · Gambits and ratings
Side 2 · The clock and history
That is the whole demo.
A real guide is this, for a whole course: more topics, more labs, a bigger bank of questions and the same hints and worked solutions. The first section of every guide is free to preview in the catalog, and a course costs $5 CAD.