Backgammon-based table game or electronic game

Abstract

A wagering table game based on backgammon is played in which a player wagers on outcomes against a dealer. A backgammon-type board is provided to at least one player that has fewer than 24 position points on the board. Each of the player and dealer are provided with fewer than 15 game pieces. Randomly generated numbers are used to allow players to move player pieces and cause the dealer to move pieces according to fixed rules. A player places a wager against a paytable, odds are paid out on player winning events, and player winning events are determined by removal of all player game pieces from the game board by movement of player game pieces on the game board.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to the field of wagering, casino table games and electronic format wagering systems.

2. Background of the Art

Table games, such as Blackjack, Baccarat, Craps, roulette and 3-Card Poker™ games are in widespread use in the gaming industry. The format of a table game is that there are multiple independent players playing against a house or dealer using a fixed strategy. The players have choices, such as wager placement, amounts of wagers, and whether to hit or stand in Blackjack, but the dealer plays a fixed strategy (e.g., in Blackjack, hitting on 16 and standing on 17 and above; and in Baccarat taking a single card when a hand count is below a minimum count). Another aspect of a table game is that it naturally favors the house. In other words, when a player makes optimal decisions, the expected return is only between 95 and 99% of the bet, such that a player may gather short term gains, but long range advantage always accrues to the house. Finally, Table games must be simple to learn, and are preferably based on familiar games.

Backgammon is considered one of the oldest games known to mankind. The game is in broad use worldwide, including Europe and the Middle East, where it is also known as Tavli or Shesh-Pesh. It is the an aspect of this invention to provide a table game based on Backgammon, which meets the four criteria of being familiar, provides limited choices to the player while the dealer uses a fixed strategy, provides expected returns in the desired range, and is simple to learn.

SUMMARY OF THE INVENTION

A wagering table game based on backgammon-type game is played on a physical format, electronic format, or mixed physical and electronic format game board. A player wagers on outcomes against a dealer. A backgammon-type board is provided to at least one player that has fewer than 24 position points on the board. Each of the player and dealer are provided with fewer than 15 game pieces. Randomly generated numbers are used to allow players to move player pieces and cause the dealer to move pieces according to fixed rules. A player places a wager against a paytable, odds are paid out on player winning events, and player winning events are determined by removal of all player game pieces from the game board by movement of player game pieces on the game board.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows five hand signals that can be used in a Nannon® wagering game or be icons on electronic game signals indicating the moves designated by the hand signals.

FIG. 2 shows a single position game layout on a table position for a Nannon® game as would be presented on a casino table felt.

DETAILED DESCRIPTION OF THE INVENTION

The Nannon® game is a simplified version of backgammon which uses a fewer number of checkers per side instead of the traditional 15 checkers used in backgammon. The Nannon® game also uses a shorter board of 6-10, instead of 24 spaces. In order to get the blocking and turnabout effects traditional in backgammon, which requires 12 checkers stacked up on six points, the Nannon® game uses adjacency of single checkers to form a block. Thus if an opponent has a checker behind two adjacent checkers, the opponent would miss a turn 33% of the time, and behind three adjacent checkers, an opponent would miss a turn 50% of the time. The Nannon® game further simplifies the underlying Backgammon game by using a single die instead of two dice, and has no complex rules for bearing off game pieces. In practice, it only takes only about 2 minutes to learn to play the Nannon® game, even though more time is likely to be needed to develop a good strategy.

The play of the Nannon® game proceeds that players take turns rolling a die, and if possible, choosing a checker to advance the amount shown on the die. The player may have a choice of moves, which require a strategic decision, or the move may be forced (only a single move is possible), or there may be no legal move so the player misses a turn. The goal of the Nannon® game is to get checkers off the opposite side of the board. Once a single checker is left, the moves are dictated by the dice rolls, and it becomes a game of probabilities and pure luck.

There are unique configurations and specifically definable events in the play of our Nannon® game, especially where one player has two checkers playing against an opponent with one checker. The one checker opponent is at an advantage towards winning, but every move is dictated by the die. The two checker player usually gets to make a few strategic decisions before the game reverts to pure luck. There are 198 positions possible, of which 111 positions have decisions to be made by the two checker player on at least one dice roll. We believe this strategic knowledge to be of the same scale as blackjack or video poker.

This configuration of two checkers against one checker meets the general criteria of providing a player with strategic choices while having a dealer use a fixed strategy—in this case moves forced by the die roll. Thus there are 3 of the 4 criteria, and are left to find configurations of this game which will provide the proper return. By default and similarity to Blackjack, it is assumed the player moves first.

Calculating the Returns.

It is known to people familiar with the field of Machine Learning that Backgammon games are Markov games, and in 1957 Bellman proved the existence of an optimal value array for such Markov games, together with an iterative algorithm which could converge to the values (R. Bellman. Dynamic Programming. Princeton University Press, Princeton, N.J., 1957). This “value iteration” algorithm starts by labeling the endgame positions with the payoffs and then updating the values for all non-ending positions using a single ply look-ahead now known as “expectimax.” The result of this iterative process is an array of expected payoff values for every possible position in the game, assuming both players make optimal moves going forward. This value array provides a “greedy” optimal move strategy by supplying the information to choose the move among alternatives which has maximum payoff.

There are 3 variables which are involved in calculating returns. The first variable is the starting position of the 3 checkers. On a six point board, these are indicated by numbers between 0 and 7. We applied this technique to a starting position of 1 & 2 versus 6, and found that the dealer would win 80% of the time, which does not provide a player with fair odds or much fun. However, while observing optimal games, we discovered two important things about the two-versus-one game. First that when the player wins, the dealer's checker is always on one of the 6 board positions, providing the player with a “lead” of 1 through 6, and second that the percentage of games where the dealer ended on the 1, 2, 3 . . . 6 point was monotonically shrinking. This is shown in Table 1 for the 1 & 2 versus 6 game:

Lead     Likelihood
Win by 1 0.0592
Win by 2 0.0448
Win by 3 0.039
Win by 4 0.0298
Win by 5 0.0228
Win by 6 0.0078
% lose   80%
% win    20%

We realized that by increasing the payout as the lead was greater, we could solve the problem of providing a reasonable return. For example, in the 1-2 versus 6 game, if the payout was {2,3,4,5,6,7}, the player could win up to 7× his bet, but this would be very rare, less than 1% of the time. Table 2 shows that calculating the return using odds in the payout raises the return from 20% to 95.4% which is in the desired range.

	Lead	Likelihood	return	Equity
	Win by 1	0.0592	3	0.1776
	Win by 2	0.0448	4	0.1792
	Win by 3	0.039	5	0.195
	Win by 4	0.0298	6	0.1788
	Win by 5	0.0228	7	0.1596
	Win by 6	0.0078	8	0.0624
	% lose	80%	0	0
	% win	20%	return:	95.26%

The second variable is the payout table, and many tables are possible, such as {1,2,3,4,5,6}, {1,1,2,2,3,3} or {1,1,2,3,5,8}. However, with each different payout table, the optimal strategy will change, so we had to derive the value array anew for each payout table. Furthermore, once the values were computed, the starting position would determine the player return. After calculating the value array, we searched for starting positions which provided a return of between 90% and 102%.

There were only a few such positions. One position stood out for the natural {1,2,3,4,5,6} payout, namely that a starting position of 1-3 versus 6 yielded a return of 98.6%, and 27% wins, and we chose this as our de facto standard for the backgammon table game, however, our invention is not limited to a six point board or this one starting position.

In order to generate more starting positions to provide flexibility to a house, we added a third variable—limiting the dealer's first roll to 1-5 or 1-4 by modifying the expectimax algorithm. We found a good number of payouts, starting conditions, and first roll limits which provide our invention with the fourth criteria, of providing a return which gives the house an edge. Forty of these are shown in table 3, sorted by return from 90% to 102%. (In order to generate interest in the game, the casino might use a >100% payoff for an introductory period.)

	Player		Dealer
Payoff Table	Start	Dealer	First Roll	Return
1	1	1	2	2	5	1	3	6	1-4	90.8%
1	1	2	2	2	25	1	3	5	1-4	91.3%
2	3	4	5	6	7	0	1	7	1-4	91.5%
1	1	1	2	2	4	1	2	7	1-4	92.0%
1	2	4	8	16	32	0	2	6	1-6	92.0%
1	2	4	8	16	32	1	2	5	1-6	92.2%
2	3	4	5	6	7	0	2	6	1-4	93.1%
1	1	2	2	2	25	0	3	7	1-6	93.6%
1	2	3	4	5	6	2	3	5	1-5	93.7%
1	1	2	2	2	25	0	2	7	1-4	94.2%
1	1	2	2	2	25	2	3	6	1-6	94.2%
0	1	2	3	4	5	1	2	7	1-5	94.4%
1	2	3	4	5	6	0	2	7	1-4	94.7%
1	1	3	3	5	5	1	2	6	1-4	94.9%
1	1	1	2	2	5	1	2	7	1-4	95.3%
2	3	4	5	6	7	1	2	6	1-6	95.4%
1	1	2	2	2	25	1	2	6	1-4	95.4%
1	2	3	5	10	25	2	3	5	1-6	95.6%
1	2	4	8	16	32	0	3	5	1-4	95.9%
1	1	1	2	2	4	1	3	7	1-6	96.3%
1	1	3	3	5	5	1	2	7	1-6	96.6%
1	1	2	3	5	8	2	3	5	1-4	96.7%
1	2	3	5	10	25	0	1	7	1-4	97.3%
1	2	3	4	5	6	1	3	6	1-6	98.6%
1	2	3	5	10	25	0	3	6	1-5	98.7%
1	1	2	3	5	8	1	2	7	1-6	98.8%
1	1	3	3	5	5	2	3	5	1-4	98.9%
1	1	3	3	5	5	0	3	7	1-4	99.6%
1	1	2	3	5	8	0	3	7	1-4	99.8%
1	2	3	4	5	6	1	3	5	1-4	99.8%
1	1	1	2	2	5	1	3	7	1-6	100.2%
1	1	2	2	2	25	2	3	5	1-4	100.5%
1	1	2	3	5	8	2	3	6	1-6	100.6%
1	1	1	2	2	4	2	3	6	1-4	100.6%
1	2	3	5	10	25	0	2	6	1-4	100.7%
0	1	2	3	4	5	2	3	6	1-5	100.8%
1	2	3	4	5	6	0	3	7	1-5	101.2%
2	3	4	5	6	7	2	3	5	1-6	101.6%
2	3	4	5	6	7	0	2	7	1-6	101.7%
1	1	1	2	2	5	2	3	6	1-4	101.7%
1	1	2	2	2	25	1	3	6	1-5	101.9%

Security Concerns

There are 2 major security concerns in a table game setup where a dealer plays against several players.

First is the fairness of the dice. Dice can be loaded, exchanged, damaged, which affect the outcome. Instead of dice, the table game could use 7 decks of custom playing cards with dice pictures, automatically shuffled and dealt as needed from a shoe as in other table games. Alternatively an electronic random element (RANDOM NUMBER GENERATOR) could be used, or even a dice Popper, as in the Trouble® game.

Secondly, if the players can move their hands across the table, they might change a bet in the middle of a game, or move a piece into a better position. Thus the dealer has to be the only person allowed to touch the checkers and the dice. Electronic wagering systems that are available on casino table games are useful in avoiding that issue.

To ameliorate this security concern, the players keep their hands off the table, and communicate their decisions with words and/or hand gestures. There are 5 such decisions which can be kept unambiguous with hand signals similar to hit and stay in Blackjack. These are FRONT, BACK, HIT, BLOCK, and RUN.

FRONT and BACK directly indicate which checker to move, the forwardmost (FRONT) or rearmost (BACK) checker. HIT indicates that one of the player's checkers can land on the dealer's checker, sending it off the board or to a restart position. RUN means to bear off with a checker, and BLOCK means to move the rear checker adjacent to the front checker into a block formation, potentially causing the dealer a missed turn. Use of these signals is not essential, except for indicating which checker to move. For example, by moving the BACK checker one space, a BLOCK may be automatically formed, without making a signal that a BLOCK was intended.

FIG. 1 shows hand signals for these 5 decisions.

FIG. 2 shows a layout of a table cloth for the game against 4 players. Each position has a board and a betting area.

A general and detailed description of play according to the present technology can include a method of playing a wagering table game based on backgammon in which a player wagers on outcomes against a dealer. A backgammon-type board with points or positions is provided to at least one player that has fewer than 24 position points on the board. Each of the player and dealer are provided with fewer than 15 game pieces. Randomly generated numbers are used to allow players to move player pieces and cause the dealer to move pieces according to fixed rules, very similar to those of backgammon, with the differences detailed above and below. A player places a wager against a paytable, odds are paid out on player winning events, and player winning events are determined by removal of all player game pieces from the game board by movement of player game pieces on the game board. The game board preferably has fewer than 10 position points on the game board and each of the player and dealer have fewer than 4 game pieces, for example, the game board has 6 position points on the game board and each of the player and dealer have fewer than 3 game pieces and payout odds are determined by dealer game piece positions at a time when a player removes all player game pieces from the game board or based on the dealer's game piece(s) initial starting position relative to the player's game pieces starting position. The preferred wagering game method is where the player has two game pieces and the dealer has a single game piece. The two player pieces may begin the game at one end of the game board and the single dealer piece begins the game at another end of the game board. The payout odds may be determined by dealer game piece positions at a time when a player removes all player game pieces from the game board. The wagering game may be played wherein the player's game pieces and dealer game piece are initially positioned on the game board by a random assignment of game pieces within a limited range of available game piece positions for player game pieces and the dealer game piece. The wagering game method may be played wherein the player's game pieces and dealer game piece are initially positioned on the game board at positions other than only extreme end positions on the game board. The wagering game method may be played wherein a dealer's first available random number used in play of the game is limited in size to a range of numbers less than a number that can move the dealer's single game piece off the game board.

Another way of describing a method of playing a wagering game on a gaming table or on an electronic gaming system can be presented as follows:

• • a) providing a game board to a player, the game board having fewer than 10 available positions on the game board for moving game pieces;
  • b) the player placing a wager against a paytable;
  • c) the player and a dealer receiving fewer than four game pieces each on the game board;
  • d) the player and the dealer alternatively receiving random numbers that are used to move respective game pieces from a starting position on the game board towards removal of game pieces from the game board;
  • e) receiving random numbers until a first of the player and the dealer have removed all of their respective game pieces from the game board; and
  • f) resolving the player's wager against the paytable.
    The method game board may be a physical game board or a virtual on a casino gaming table wherein:
  • the player is given two game pieces;
  • the dealer is given a single game piece;
  • the player game pieces are placed within a first three points on a game board having six points thereon;
  • the dealer's game piece is placed within a set of last three points on the game board having six points thereon;
  • random numbers are provided by a single die or a single value provided by a random number generator; and
  • the player's wager is resolved as a win when all of the player's game pieces are removed from the game board before the dealer's game piece is removed from the game board.

The method can be played, for example, wherein a dealer's first available random number used in play of the game is limited in size to a range of numbers less than a number that can move the dealer's single game piece off the game board. Adjacent game pieces may form blockades against movement of game pieces. Initial positioning of the player's and dealer's game pieces at the start of the game may determine payout awards on the paytable for a player winning outcome. The dealer's game piece position when all of a player's game pieces have been removed from the game board may determine payout awards on the paytable for a player's winning outcome. The game board may be an electronic game board with a processor and video display, random numbers are generated by a random number generator in the processor, and wagers are placed electronically and received by the processor. The method may be strategically played wherein the player's two game pieces are position at points 1 and 3 on the game board and the dealer's game piece is positioned at point 6 on the game board.

In an electronic gaming system, many different combinations or alternatives in construction may be played. Player input can be through buttons, touchscreens, personal data entry systems (PDA's, cell phones, etc.), player terminals on a communal system, stand-alone units, and the like. Gaming systems will have at least one processor and information display to provide information to at least the player on wagers, outcomes and game progression. Rules will be controlled or executed (e.g., dealer's will have all moves controlled by the rules without exercise of judgment) and accounting of wagers and outcomes will be determined by a processor.

Claims

1. A method of playing a wagering table game based on backgammon in which a player wagers on outcomes against a dealer, wherein: wherein a player places a wager against a paytable, odds are paid out on player winning events, and player winning events are determined by removal of all player game pieces from the game board by movement of player game pieces on the game board.

a backgammon board is provided to at least one player that has fewer than 24 position points on the board;

each of the player and dealer are provided with fewer than 15 game pieces;

randomly generated numbers are used to allow players to move player pieces and cause the dealer to move pieces according to fixed rules;

2. The wagering game method of claim 1 wherein the game board has fewer than 10 position points on the game board and each of the player and dealer have fewer than 4 game pieces.

3. The wagering game method of claim 1 wherein the game board has 6 position points on the game board and each of the player and dealer have fewer than 3 game pieces and payout odds are determined by dealer game piece positions at a time when a player removes all player game pieces from the game board.

4. The wagering game method of claim 3 wherein the player has two game pieces and the dealer has a single game piece.

5. The wagering game method of claim 4 wherein the two player pieces begin the game at one end of the game board and the single dealer piece begins the game at another end of the game board.

6. The wagering game method of claim 3 wherein payout odds are determined by dealer game piece positions at a time when a player removes all player game pieces from the game board.

7. The wagering game method of claim 4 wherein payout odds are determined by dealer game piece positions at a time when a player removes all player game pieces from the game board.

8. The wagering game method of claim 5 wherein payout odds are determined by dealer game piece positions at a time when a player removes all player game pieces from the game board.

9. The wagering game method of claim 6 wherein the player's game pieces and dealer game piece are initially positioned on the game board by a random assignment of game pieces within a limited range of available game piece positions for player game pieces and the dealer game piece.

10. The wagering game method of claim 6 wherein the player's game pieces and dealer game piece are initially positioned on the game board at positions other than only extreme end positions on the game board.

11. The wagering game method of claim 4 wherein a dealer's first available random number used in play of the game is Limited in size to a range of numbers less than a number that can move the dealer's single game piece off the game board.

12. The wagering game method of claim 7 wherein a dealer's first available random number used in play of the game is limited in size to a range of numbers less than a number that can move the dealer's single game piece off the game board.

13. A method of playing a wagering game on a gaming table or on an electronic gaming system comprising:

a) providing a game board to a player, the game board having fewer than 10 available positions on the game board for moving game pieces;

b) the player placing a wager against a paytable;

c) the player and a dealer receiving fewer than four game pieces each on the game board;

d) the player and the dealer alternatively receiving random numbers that are used to move respective game pieces from a starting position on the game board towards removal of game pieces from the game board;

e) receiving random numbers until a first of the player and the dealer have removed all of their respective game pieces from the game board; and

f) resolving the player's wager against the paytable.

14. The method of claim 13 wherein the game board is a physical game board on a casino gaming table;

the player is given two game pieces;

the dealer is given a single game piece;

the player game pieces are placed within a first three points on a game board having six points thereon;

the dealer's game piece is placed within a set of last three points on the game board having six points thereon;

random numbers are provided by a single die; and

the player's wager is resolved as a win when all of the player's game pieces are removed from the game board before the dealer's game piece is removed from the game board.

15. The method of claim 14 wherein a dealer's first available random number used in play of the game is limited in size to a range of numbers less than a number that can move the dealer's single game piece off the game board.

16. The method of claim 14 wherein adjacent game pieces form blockades against movement of game pieces.

17. The method of claim 14 wherein initial positioning of the dealer's game piece at the start of the game determines payout awards on the paytable for a player winning outcome.

18. The method of claim 14 wherein the dealer's game piece position when all players' game pieces have been removed from the game board determines payout awards on the paytable for a player's winning outcome.

19. The method of claim 14 wherein the game board is an electronic game board with a processor and video display, random numbers are generated by a random number generator in the processor, and wagers are placed electronically and received by the processor.

20. The method of claim 14 wherein the player's two game pieces are position at points 1 and 3 on the game board and the dealer's game piece is positioned at point 6 on the game board.