Ranking Games that have Competitiveness-based Strategies
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Ranking Games that have Competitiveness- based Strategies Leslie Goldberg, Paul Goldberg, Piotr Krysta and Carmine Ventre University of Liverpool
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Ranking Games that have Competitiveness-based Strategies. Leslie Goldberg, Paul Goldberg, Piotr Krysta and Carmine Ventre University of Liverpool. Ἐν ἀρχῇ ἦν ὁ ἀρρεψία Nash , καὶ ὁ ἀρρεψία Nash ἦν πρὸς τὸν οἰκονόμος. - PowerPoint PPT Presentation
Transcript of Ranking Games that have Competitiveness-based Strategies
Ranking Games that have Competitiveness-based Strategies
Leslie Goldberg, Paul Goldberg, Piotr Krysta and Carmine Ventre
University of Liverpool
Ἐν ἀρχῇ ἦν ὁ ἀρρεψία Nash, καὶ ὁ ἀρρεψία Nash ἦν πρὸς τὸν οἰκονόμος...
In the beginning was the Nash equilibrium, and the Nash equilibrium was with Economists…
... but then in the summer of 2005... ...Daskalakis, Goldberg & Papadimitriou show
that computing NEs is “hard” (in terms of PPAD) for graphical games
Later, [DP, Chen & Deng] show “hardness” for 3-player games and CD show “hardness” for 2-player games!
Q: Is NE a “meaningful” concept? “If your laptop can't find it, neither can the market.”
Kamal Jain. A1: Define interesting classes of games (ie,
describing the world) for which it is A2: Compute efficiently approximate NEs
Ἐν ἀρχῇ ἦν ὁ ἀρρεψία Nash, καὶ ὁ ἀρρεψία Nash ἦν πρὸς τὸν οἰκονόμος, καὶ μπορείἀκμήν ἐγγίων ἀρρεψία καὶ / ή ἀξιόλογος ἀστροθετέω των ἄεθλος
In the beginning was the Nash equilibrium, and the Nash equilibrium was with Economists, and it may still be for approximate equilibria and/or an interesting class of games
Every morning in Africa...... a Gazelle wakes up. It knows it must run faster than the fastest lion or it will be killed. Every morning a Lion wakes up. It knows it must outrun the slowest Gazelle or it will starve to death. It doesn't matter whether you are a Lion or a Gazelle... when the sun comes up, you'd better be running
0 mph
0 mph
25 mph 50 mph
25 mph
50 mph
Ranking games
1st
2nd
1st
2nd
A1: Define interesting classes of games (describing the world) for which NE is “meaningful”
Ranking games describe the world but NE is not “meaningful” for them (ie, these games are “hard”). [Brandt, Fischer, Harrenstein & Shoham, 2009]
Competitiveness-based ranking games0 mph
0 mph
25 mph 50 mph
25 mph
50 mph
Increasing effort
Increasing effort
cost (effort)
return (speed)
Aside note: Returns allow compact representation of these games
A1
A2
Our algorithmic results
# players # prizes # actions Result
return-symmetric games
O(1) O(1) any PTAS
any any O(1) PTAS
O(1) 1 any FPTAS
any any 2 Exact pure
tie-free games 2 2 any Exact
linear-prize games any # players any Exact
A (F)PTAS computes an Ɛ-NE in time polynomial in the input (and 1/Ɛ)
1
2 4 6 8 10
3
5
7
9
Games without ties Return values are all different
E.g., no two players ranked first, Google page rank Algorithm to find NEs of any 2-player such game
1 wins
2 wins
1. The support of a NE is a prefix of the strategies available to a player
2. There is a polynomial number of possible supports3. It is well known that once having the support we can
efficiently solve a 2-player game (essentially LP)
Games without ties (further results) Characterization of NEs for games with a single
prize: “One player has expected payoff positive, all the others have expected payoff 0.”
Games without ties and single prize can be solved in polytime given the knowledge of the support
Reduction to polymatrix games [DP09] when prizes are linear (rank j has a prize a-jb) Polymatrix games and thus linear-prize ranking games
are solvable in polytime [DP09]
Return-symmetric games (RSGs) All players have n actions, all with the same return
while cost-per-action is player specific E.g., lion-gazelle game
Actions’ returns: all speeds in [0,50] mph Effort for speed s is animal/player-dependant
NEs of these games can be studied wlog* for our class of ranking games
cost2(r) = cost2(r’’)
r’ r’’
r
r’
r’’
r’ < r < r’’
r
r’
r’’
r’ r’’r
* A game with O(1) actions can be reduced to a game with a polynomial number of actions
PTAS for RSGs with O(1) players 1. Round down each cost (normalized to [0,1]) to the nearest integer multiple of Ɛ
2. Eliminate dominated strategies
3. Brute force search for an Ɛ-NE of the reduced game using discretized probability vectors (prob’s are integer multiple of δ) (in time (k+1)(#players/δ))
1
n
1n
10
Ɛ=1/k, δ=Ɛ/(k+1) for k in NRegret of 3Ɛ
regret of Ɛ
regret of 2Ɛ
After step 2 each player has only k+1 strategies
polytime
FPTAS for RSGs, O(1) players and single prize worth 1
j
1 j-1 j j+1 n
winsharelose
)(21
1221 jCostxx jji ij
Definition of Ɛ-NE:
1.x’s are probability distribution2.
0max
0...max111
11
12
11
11
jijij
jijij
x
xxx
01 ji ix11j
21j
11j
FPTAS: left-to-right 21 , nn 2
111
21
11
21
11
21
11 ,,,,,,, xx 2
212
22
12
22
12
22
12 ,,,,,,, xx
2,3,5,,2,3,4,2,is a collection of vectors of admissibile values that are multiple of Ɛ,e.g.,
…
…
Discard : sequences whose first 8 values are different than last 8 values of previous sequences
FPTAS: right-to-left
Output the x’s in
21
11
21
11
21
11
21
11 ,,,,,,, xx 2
212
22
12
22
12
22
12 ,,,,,,, xx
…
21 , nn …
Overall regret of
Ɛ
= O(1/Ɛ9) FPTAS
Conclusions
Introduction of ranking games with competitiveness-based strategies Interesting games (describing real life) Encouraging initial positive results (wrt both A1, A2)
Work in progress: FPTAS works for many prizes Open problems
What is the hardness of these games? Related to the unknown hardness of anonymous games
Polytime algorithms for 2-player RSGs?
