Proof

Previously... Art Communicates Subjective, Qualitative Science Establishes Facts Objective, Quantitative, Reproducible Experiments Proofs Game Design Unifies Both

Experiments Design Measure appropriate value Avoid bias Analysis Sample (m, s) vs. population (μ, σ) statistics Confidence that μ is within m ± t*s Percentage of population within μ ± k*σ

Example

The Magistrates ‘A’-level mechanics analysis Brief summary of rules (2 pages w/pics) Introduces terminology ‘feature’, ‘completed’, ‘follower’ Uses technical language ‘finite’, ‘score’, ‘connected’, ‘adjacent’

Analysis Techniques Define Terms ‘trivially bad’, ‘strategic complexity’ Experimental Evidence Section 3.5: Play Time Proof Section using math Section using text Argument & Discussion Section 3.2: Random Tile Distribution Section 3.6: Adjusted Point Awards

Experimental Evidence Read Section 3.5 Outliers What does t = mean? m-ts < μ < m+ts Where did equation for s Δ come from? Variance of a sum or difference is the sum of the variances (CG ch. 9) Why prove that 0 < μ Δ with high probability? If the difference of the before and after sample times is likely greater than zero, then it is likely that the after time truly is less than the before time; this proves that the mod really makes the game shorter.

Proof Example Read section of The Magistrates Uses no math symbols! Proves a (weak) bound on the change in complexity Actual case was too hard, so bounds are the best we can do Follows up in with arguments (but not proof) for a stronger result

Proof Proof or Derivation Experimental Evidence (Persuasive Argument) Fact Opinion Historical Observation

Vocabulary Theorem: statement you’re trying to prove. e.g., “Players are more likely to draw city tiles than road tiles in Carcassonne” “Checkers is a fair game” Axiom: unproven theorems that we take as given parts of our mathematics. e.g., If a > b and b > c, then a > c Assumption: something that you’ll pretend is true throughout the proof; if it turns out to not be true then your proof does not hold. Proof: Sequence of individually “obvious” statements that reduce the theorem statement to a known true statement.

Techniques Some common proof structures: Exhaustion Induction Derivation Contradiction Reduction Methods for simplifying complex cases: Bounding Approximation Game problems are so complex that we frequently have to rely on the bold techniques

Exhaustion Go through every possible case and show that it follows your theorem This does not work for large or variable-size problems! Examples: “There are more city tiles than road tiles in Carcassonne” “No hex in Through the Desert is more than 2 hexes from a water feature”

Exercise

Induction Induction Axiom: A theorem holds for n > k if it can be proven for n = k and can be proven for n = m + 1 by assuming that it is true for n = m.

Exercise Theorem: “The expected value of 1dn is (n+1)/2 for n >= 1” Approach: 1. Prove that E[1d1] = (1+1)/2 2. Assume that E[1dm] = (m+1)/2 and prove that E[1d(m+1)] = (m+1+1)/2

Induction Exercise Theorem: “The expected value of 1dn is (n+1)/2 for n >= 1” a) E[1d1] = 1 * P(x = 1) = 1 * 1 = 1 i.e., definition of expected value 1 = (1+1)/2, therefore the theorem holds for n = 1 b) Assume E[1dm] = (m+1)/2 E[1d(m+1)] = E[1dm] * m/(m+1) + (m+1)*P(x = m+1) i.e., the summation for m+1 contains only one more term = (m+1)/2 * m/(m+1) + (m+1)*1/(m+1) = m/2 + 1 = (m+1) + 1) / 2, therefore the theorem holds for n = m + 1 By induction, the theorem is true.

Derivation In some cases we want to directly derive the solution rather than proving properties about it. Examples: Number n-player rounds in a Carcassonne game = 71 / n Minimum round-trip latency using UDP: t = 2 * [ encode time + transmit time] = 2 * [ number of bits / (bits/sec encode) + distance / (propagation rate)]

Dirty Secret Run the derivation backwards and it is a proof.

Example The Magistrates, section : Derivation Example 3.3.2: Proof Example Induction using derivation for the 2nd part

Contradiction Prove that the opposite of the statement is false

Reduction Show that this is equivalent to a previously solved problem Example: Magic 15 Game

“Magic 15” 1. Place cards with numbers face up 2. Two players take turns drawing one card each 3. First player with three cards that sum to exactly 15 wins Theorem: Magic 15 is fair and always ends in a draw with perfect play.

Proof by Reduction Theorem: Magic 15 is fair and always ends in a draw with perfect play. Proof: Arrange the 9 cards in the magic square shown on the right Drawing a card is equivalent to “marking” one grid cell of the magic square Only rows, columns, and diagonals sum to exactly 15, so winning requires marking one of those The above two rules describe Tic-Tac-Toe, so Magic 15 reduces to Tic-Tac-Toe Tic-Tac-Toe is fair and always ends in a draw.

Bounding Solve two simpler problems, whose results must be bigger and smaller, thus proving that the result for the original problem lies between them. Example: The Magistrates section This is weaker than finding an exact solution

Approximation Solve a simpler problem and then prove (or argue) that the difference in result from the simplification is bounded. This is generally weaker than bounding, and is at best equivalent to it Example: The Magistrates, section 3.3 “For simplicity, assume that all tiles are unique...our evaluation of U will be an upper bound.”