Penney's game, named after its inventor Walter Penney, is a binary (head/tail) sequence generating game between two players. Player A selects a sequence of heads and tails (of length 3 or larger), and shows this sequence to player B. Player B then selects another sequence of heads and tails of the same length. Subsequently, a fair coin is tossed until either player A's or player B's sequence appears as a consecutive subsequence of the coin toss outcomes. The player whose sequence appears first wins.
A possible sequence in Penney's game: heads, tails, headsGraphs of best responses for Penney's games of sequence lengths 3 and 4 – each sequence is dominated by the sequence pointing to it with the given probability (italics) or odds (normal text)[1]
Provided sequences of at least length three are used, the second player (B) has an edge over the starting player (A). This is because the game is nontransitive such that for any given sequence of length three or longer one can find another sequence that has higher probability of occurring first.
For the three-bit sequence game, the second player can optimize their odds by choosing sequences according to:
1st player's choice
2nd player's choice
Odds in favour of 2nd player
HHH
THH
7 to 1
HHT
THH
3 to 1
HTH
HHT
2 to 1
HTT
HHT
2 to 1
THH
TTH
2 to 1
THT
TTH
2 to 1
TTH
HTT
3 to 1
TTT
HTT
7 to 1
An easy way to remember the sequence is for the second player to start with the opposite of the middle choice of the first player, then follow it with the first player's first two choices.
So for the first player's choice of 1-2-3
the second player must choose (not-2)-1-2
where (not-2) is the opposite of the second choice of the first player.
An intuitive explanation for this result is that in any case that the sequence is not immediately the first player's choice, the chances for the first player getting their sequence-beginning, the opening two choices, are usually the chance that the second player will be getting their full sequence. So the second player will most likely "finish before" the first player.
Strategy for more than three bitsedit
The optimal strategy for the first player (for any length of the sequence no less than 4) was found by J.A. Csirik (See References). It is to choose HTTTT.....TTTHH ( T's) in which case the second player's maximal odds of winning is .
Variation with playing cardsedit
One suggested variation on Penney's Game uses a pack of ordinary playing cards. The Humble-Nishiyama Randomness Game follows the same format using Red and Black cards, instead of Heads and Tails.[2][3] The game is played as follows. At the start of a game each player decides on their three colour sequence for the whole game. The cards are then turned over one at a time and placed in a line, until one of the chosen triples appears. The winning player takes the upturned cards, having won that "trick". The game continues with the rest of the unused cards, with players collecting tricks as their triples come up, until all the cards in the pack have been used. The winner of the game is the player that has won the most tricks. An average game will consist of around 7 "tricks". As this card-based version is quite similar to multiple repetitions of the original coin game, the second player's advantage is greatly amplified. The probabilities are slightly different because the odds for each flip of a coin are independent while the odds of drawing a red or black card each time is dependent on previous draws. Note that HHT is a 2:1 favorite over HTH and HTT but the odds are different for BBR over BRB and BRR.
Below are approximate probabilities of the outcomes for each strategy based on computer simulations:[4]
1st player's choice
2nd player's choice
Probability 1st player wins
Probability 2nd player wins
Probability of a draw
BBB
RBB
0.11%
99.49%
0.40%
BBR
RBB
2.62%
93.54%
3.84%
BRB
BBR
11.61%
80.11%
8.28%
BRR
BBR
5.18%
88.29%
6.53%
RBB
RRB
5.18%
88.29%
6.53%
RBR
RRB
11.61%
80.11%
8.28%
RRB
BRR
2.62%
93.54%
3.84%
RRR
BRR
0.11%
99.49%
0.40%
If the game is ended after the first trick, there is a negligible chance of a draw. The odds of the second player winning in such a game appear in the table below.
1st player's choice
2nd player's choice
Odds in favour of 2nd player
BBB
RBB
7.50 to 1
BBR
RBB
3.08 to 1
BRB
BBR
1.99 to 1
BRR
BBR
2.04 to 1
RBB
RRB
2.04 to 1
RBR
RRB
1.99 to 1
RRB
BRR
3.08 to 1
RRR
BRR
7.50 to 1
Variation with a Roulette wheeledit
Recently Robert W. Vallin, and later Vallin and Aaron M. Montgomery, presented results with Penney's Game as it applies to (American) roulette with Players choosing Red/Black rather than Heads/Tails. In this situation the probability of the ball landing on red or black is 9/19 and the remaining 1/19 is the chance the ball lands on green for the numbers 0 and 00. There are various ways to interpret green: (1) as a "wild card" so that BGR can be read at Black, Black, Red and Black, Red, Red, (2) as a do-over, the game stops when green appears and restarts with the next spin, (3) as just itself with not extra interpretation. Results have been worked out for odds and wait times.[5]
^"Best response strings for strings of length 3. | Download Scientific Diagram". Retrieved 29 March 2023.
^Winning Odds by Yutaka Nishiyama and Steve Humble
^Humble-Nishiyama Randomness Game - A New Variation on Penney’s Coin Game on CiteSeer
^Results are broadly in line with those in Steve Humble and Yutaka Nishiyama, Humble-Nishiyama Randomness Game Mathematics Today August 2010 p 143 - A new variation on Penney’s Coin Game [1] 24 September 2015 at the Wayback Machine
^Jennifer Beineke; Jason Rosenhouse; Robert W. Vallin (5 September 2017). The Mathematics of Various Entertaining Subjects: Research in Games, Graphs, Counting, and Complexity, Volume 2. Princeton: Princeton University Press. ISBN9780691171920.
Walter Penney, Journal of Recreational Mathematics, October 1969, p. 241.
Martin Gardner, "Time Travel and Other Mathematical Bewilderments", W. H. Freeman, 1988.
L.J. Guibas and A.M. Odlyzko, "String Overlaps, Pattern Matching, and Nontransitive Games", Journal of Combinatorial Theory, Series A. Volume 30, Issue 2, (1981), pp 183–208.
S. Humble & Y. Nishiyama, "Humble-Nishiyama Randomness Game - A New Variation on Penney's Coin Game", IMA Mathematics Today. Vol 46, No. 4, August 2010, pp 194–195.
Steve Humble & Yutaka Nishiyama, "Winning Odds", Plus Magazine, Issue 55, June 2010.
Yutaka Nishiyama, Pattern Matching Probabilities and Paradoxes as a New Variation on Penney’s Coin Game, International Journal of Pure and Applied Mathematics, Vol.59, No.3, 2010, 357-366.
Ed Pegg, Jr., "How to Win at Coin Flipping", Wolfram Blog, 30 November 2010.
Robert W. Vallin "A sequence game on a roulette wheel", The Mathematics of Very Entertaining Subjects: Research in Recreational Math, Volume II, Princeton University Press, (to be published in 2017)
James Brofos, "A Markov Chain Analysis of a Pattern Matching Coin Game." arXiv:1406.2212 (2014).
penney, game, named, after, inventor, walter, penney, binary, head, tail, sequence, generating, game, between, players, player, selects, sequence, heads, tails, length, larger, shows, this, sequence, player, player, then, selects, another, sequence, heads, tai. Penney s game named after its inventor Walter Penney is a binary head tail sequence generating game between two players Player A selects a sequence of heads and tails of length 3 or larger and shows this sequence to player B Player B then selects another sequence of heads and tails of the same length Subsequently a fair coin is tossed until either player A s or player B s sequence appears as a consecutive subsequence of the coin toss outcomes The player whose sequence appears first wins A possible sequence in Penney s game heads tails heads Graphs of best responses for Penney s games of sequence lengths 3 and 4 each sequence is dominated by the sequence pointing to it with the given probability italics or odds normal text 1 Provided sequences of at least length three are used the second player B has an edge over the starting player A This is because the game is nontransitive such that for any given sequence of length three or longer one can find another sequence that has higher probability of occurring first Contents 1 Analysis of the three bit game 2 Strategy for more than three bits 3 Variation with playing cards 4 Variation with a Roulette wheel 5 See also 6 References 7 External linksAnalysis of the three bit game editFor the three bit sequence game the second player can optimize their odds by choosing sequences according to 1st player s choice 2nd player s choice Odds in favour of 2nd player HHH THH 7 to 1 HHT THH 3 to 1 HTH HHT 2 to 1 HTT HHT 2 to 1 THH TTH 2 to 1 THT TTH 2 to 1 TTH HTT 3 to 1 TTT HTT 7 to 1 An easy way to remember the sequence is for the second player to start with the opposite of the middle choice of the first player then follow it with the first player s first two choices So for the first player s choice of 1 2 3 the second player must choose not 2 1 2 where not 2 is the opposite of the second choice of the first player An intuitive explanation for this result is that in any case that the sequence is not immediately the first player s choice the chances for the first player getting their sequence beginning the opening two choices are usually the chance that the second player will be getting their full sequence So the second player will most likely finish before the first player Strategy for more than three bits editThe optimal strategy for the first player for any length of the sequence no less than 4 was found by J A Csirik See References It is to choose HTTTT TTTHH k 3 displaystyle k 3 nbsp T s in which case the second player s maximal odds of winning is 2 k 1 1 2 k 2 1 displaystyle 2 k 1 1 2 k 2 1 nbsp Variation with playing cards editOne suggested variation on Penney s Game uses a pack of ordinary playing cards The Humble Nishiyama Randomness Game follows the same format using Red and Black cards instead of Heads and Tails 2 3 The game is played as follows At the start of a game each player decides on their three colour sequence for the whole game The cards are then turned over one at a time and placed in a line until one of the chosen triples appears The winning player takes the upturned cards having won that trick The game continues with the rest of the unused cards with players collecting tricks as their triples come up until all the cards in the pack have been used The winner of the game is the player that has won the most tricks An average game will consist of around 7 tricks As this card based version is quite similar to multiple repetitions of the original coin game the second player s advantage is greatly amplified The probabilities are slightly different because the odds for each flip of a coin are independent while the odds of drawing a red or black card each time is dependent on previous draws Note that HHT is a 2 1 favorite over HTH and HTT but the odds are different for BBR over BRB and BRR Below are approximate probabilities of the outcomes for each strategy based on computer simulations 4 1st player s choice 2nd player s choice Probability 1st player wins Probability 2nd player wins Probability of a draw BBB RBB 0 11 99 49 0 40 BBR RBB 2 62 93 54 3 84 BRB BBR 11 61 80 11 8 28 BRR BBR 5 18 88 29 6 53 RBB RRB 5 18 88 29 6 53 RBR RRB 11 61 80 11 8 28 RRB BRR 2 62 93 54 3 84 RRR BRR 0 11 99 49 0 40 If the game is ended after the first trick there is a negligible chance of a draw The odds of the second player winning in such a game appear in the table below 1st player s choice 2nd player s choice Odds in favour of 2nd player BBB RBB 7 50 to 1 BBR RBB 3 08 to 1 BRB BBR 1 99 to 1 BRR BBR 2 04 to 1 RBB RRB 2 04 to 1 RBR RRB 1 99 to 1 RRB BRR 3 08 to 1 RRR BRR 7 50 to 1Variation with a Roulette wheel editRecently Robert W Vallin and later Vallin and Aaron M Montgomery presented results with Penney s Game as it applies to American roulette with Players choosing Red Black rather than Heads Tails In this situation the probability of the ball landing on red or black is 9 19 and the remaining 1 19 is the chance the ball lands on green for the numbers 0 and 00 There are various ways to interpret green 1 as a wild card so that BGR can be read at Black Black Red and Black Red Red 2 as a do over the game stops when green appears and restarts with the next spin 3 as just itself with not extra interpretation Results have been worked out for odds and wait times 5 See also editNontransitive gameReferences edit Best response strings for strings of length 3 Download Scientific Diagram Retrieved 29 March 2023 Winning Odds by Yutaka Nishiyama and Steve Humble Humble Nishiyama Randomness Game A New Variation on Penney s Coin Game on CiteSeer Results are broadly in line with those in Steve Humble and Yutaka Nishiyama Humble Nishiyama Randomness Game Mathematics Today August 2010 p 143 A new variation on Penney s Coin Game 1 Archived 24 September 2015 at the Wayback Machine Jennifer Beineke Jason Rosenhouse Robert W Vallin 5 September 2017 The Mathematics of Various Entertaining Subjects Research in Games Graphs Counting and Complexity Volume 2 Princeton Princeton University Press ISBN 9780691171920 Walter Penney Journal of Recreational Mathematics October 1969 p 241 Martin Gardner Time Travel and Other Mathematical Bewilderments W H Freeman 1988 L J Guibas and A M Odlyzko String Overlaps Pattern Matching and Nontransitive Games Journal of Combinatorial Theory Series A Volume 30 Issue 2 1981 pp 183 208 Elwyn R Berlekamp John H Conway and Richard K Guy Winning Ways for your Mathematical Plays 2nd Edition Volume 4 AK Peters 2004 p 885 S Humble amp Y Nishiyama Humble Nishiyama Randomness Game A New Variation on Penney s Coin Game IMA Mathematics Today Vol 46 No 4 August 2010 pp 194 195 Steve Humble amp Yutaka Nishiyama Winning Odds Plus Magazine Issue 55 June 2010 Yutaka Nishiyama Pattern Matching Probabilities and Paradoxes as a New Variation on Penney s Coin Game International Journal of Pure and Applied Mathematics Vol 59 No 3 2010 357 366 Ed Pegg Jr How to Win at Coin Flipping Wolfram Blog 30 November 2010 J A Csirik Optimal strategy for the first player in the Penney ante game Combinatorics Probability and Computing Volume 1 Issue 4 1992 pp 311 321 Robert W Vallin A sequence game on a roulette wheel The Mathematics of Very Entertaining Subjects Research in Recreational Math Volume II Princeton University Press to be published in 2017 James Brofos A Markov Chain Analysis of a Pattern Matching Coin Game arXiv 1406 2212 2014 External links editAn online simulation of Penney s game Some variants of Penney s game Retrieved from https en wikipedia org w index php title Penney 27s game amp oldid 1186165684, wikipedia, wiki, book, books, library,
