Definitions Ahmed

Definition: A ranking function r is said to have the LimitedCollusionEffect (LCE) property if for any two arbitrary players px and py the rank of py with respect to px cannot be improved by manipulating games that px cannot control. Correct? The rank of px cannot be improved by manipulating games that px cannot control. Correct? The rank difference between px and py cannot shortened by manipulating games that neither px nor py can control. What is the definition of weakly better?

games c: under control of Alice* !c: not under control of Alice *These are the games that Alice either wins or had a chance to win.

Games under control of Aliceunder control of Bob

3 Player Example Games under control of Alice under control of Bob under control of Henry 3 game kinds: black, blue and red. black and blue are under the control of Alice provided she was not forced. red not under Alice’ control. black and red are under the control of Henry provided he is not forced. blue not under Henry’s control. blue and red are under the control of Bob provided he is not forced. black is not under control of Bob. black: 5 games blue: 1 game red: 1 game Bob not forced Bob:Alice 1:0 Bob:Henry 1:0 Alice:Henry 5:0

3 Player Example: LCE Games under control of Alice under control of Bob under control of Henry 3 game kinds: black, blue and red. black and blue are under the control of Alice provided she was not forced and lost. red not under Alice’ control. black and red are under the control of Henry provided he is not forced and lost. blue not under Henry’s control. blue and red are under the control of Bob provided he is not forced and lost. black is not under control of Bob. black: 5 games blue: 1 game red: 1 game Bob not forced Bob:Alice 1:0 Bob:Henry 1:0 Alice:Henry 5:0 px py px <= py improve for px to py <= px only possible by changing game results that px can control.

LFB A ranking function r is said to be Local Fault Based (LFB) if for any two arbitrary players px and py the relative rank it assigns to px with respect to py solely depends on the faults made by either px or py. ? Faults made by both px and py.

It is inacceptable for a ranking function to reward losing or to penalize winning. In other words, a ranking function must have a Non-Negative Regard for Winning (NNRW) and a {Non-Positive Regard for Losing (NPRL)}. That is, a player's rank cannot be worsened by an extra winning nor can it be improved by an extra loss.

For any ranking function having NNRW and NPRL, LCE is equivalent to LFB.

Neutrality A ranking function r is said to be neutral if the order it assigns to some beating function is preserved under any permutation on the players preserving the beating function.

The following theorem provides a complete characterization of the loss counting ranking function. Fault Counting is the most refined, neutral ranking function that satisfies the limited collusion effect property and has a positive regard for wins.

f satisfies the NNRW property. f satisfies the neutrality property. $\rank{f}$ f refines any ranking function satisfying LCE, NNRW, and neutrality properties.

CLRS page 374, optimal parenthesization m[i,j]=m[i,k]+m[k+1,j] + p i-1 pk pj want m[1,n] m[i,j]=min[k]{m[i,k]+m[k+1,j]+p i-1 pk pj} for i<j. HSR m[l,i,j] = depth of optimal decision tree with leaves i-1 to j-1 and l jars to break. m[l,i,j] = max(m[l-1,i,k],m[l,k+1,j) + 1 want m[l,1,n] m[l,i,j]=min[i<=k<j]{max(m[l-1,i,k],m[l,k+1,j])+1} for i<j.

CLRS page 374, optimal parenthesization m[i,j]=m[i,k]+m[k+1,j] + p i-1 pk pj want m[1,n] m[i,j]=min[k]{m[i,k]+m[k+1,j]+p i-1 pk pj} for i<j. HSR m[l,i,j] = depth of optimal decision tree with leaves i-1 to j-1 and l jars to break. m[l,i,j] = max(m[l-1,i,k],m[l,k+1,j) + 1 want m[l,1,n] m[l,i,j]=min[i<=k<j]{max(m[l-1,i,k],m[l,k+1,j])+1} for i<j.

CLRS page 374, optimal parenthesization m[i,j]=m[i,k]+m[k+1,j] + p i-1 pk pj want m[1,n] m[i,j]=min[k]{m[i,k]+m[k+1,j]+p i-1 pk pj} for i<j. HSR m[l,i,j] = depth of optimal decision tree with leaves i-1 to j-1 and l jars to break. m[l,i,j] = max(m[l-1,i,k],m[l,k+1,j) + 1 want m[l,1,n] m[l,i,j]=min[i<=k<j]{max(m[l-1,i,k],m[l,k+1,j])+1} for i<j. m[l,n]=min[1 1. m[l,n]=depth of optimal decision tree with leaves 0..n-1 and l jars to break.

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