Tennis (pencil game)

Tennis is a strategic paper-and-pencil game for two players.

regulate

The playing field consists of 7 fields, which are labeled with the numbers −3 to +3. At the beginning the ball lies on field 0. Each player has a points account with initially z. B. 50 points. Player plus tries to hit the ball on field 3, and player minus tries to hit it on field −3.

In each turn, the players simultaneously choose a natural number that must not be greater than their respective score. If you haven't used up all the points, you have to choose a positive number. Whoever names the higher number hits the ball on the opposing side or, if it is already there, one space further. For example , if Player Plus names the higher number, then the ball is hit on the next higher field, but at least on field 1. If the ball lands on field 3, Player Plus wins . For players minus the same applies in the opposite direction. If they both choose the same number, the ball position does not change. The number chosen by each player is subtracted from their score before the next move is played.

The game ends even if no player has any more points. Then plus wins if the ball is on a positive field and negative if it is on a negative field. If it is on the 0 space (which only happens if both choose the same number on each turn), the game ends in a draw.

If you play several times (until a certain point limit has been reached), you can specify that 2 profit points are awarded for knocking out over the baseline (ball on field 3 or −3), otherwise only 1 profit point.

Mathematical description

At the beginning is the ball position and the players have the score . Player 1 corresponds to Player Plus . ${\displaystyle B_{0}=0}$ ${\displaystyle i\in \{1,2\}}$ ${\displaystyle S_{i,0}=50}$

In train each player chooses an integer as his train with . The selected number reduces the player's points: . The following applies to the new ball position: ${\displaystyle t\in \mathbb {N} ^{+}}$ ${\displaystyle Z_{i,t}}$ ${\displaystyle \min(1,S_{i,t-1})\leq Z_{i,t}\leq S_{i,t-1}}$ ${\displaystyle S_{i,t}=S_{i,t-1}-Z_{i,t}}$

${\displaystyle B_{t}={\begin{cases}\max(1,B_{t-1}+1)&{\text{wenn}}\;Z_{1,t}>Z_{2,t}\\B_{t-1}&{\text{wenn}}\;Z_{1,t}=Z_{2,t}\\\min(-1,B_{t-1}-1)&{\text{wenn}}\;Z_{1,t}<Z_{2,t}\end{cases}}}$

The game ends as soon as or or . If so then player 1 wins, and if so then player 2 wins , and if so , the game ends in a tie. ${\displaystyle B_{t}<-2}$ ${\displaystyle B_{t}>2}$ ${\displaystyle S_{1,t}=S_{2,t}=0}$ ${\displaystyle B_{t}>0}$ ${\displaystyle B_{t}<0}$ ${\displaystyle B_{t}=0}$

Sample games

Course of the game according to table 1

In the first example, player 1 wins after both players run out of points (ball still in the field).

t	Player 1 turn ${\displaystyle S_{1,t}}$	Player 2 turn ${\displaystyle S_{2,t}}$	Player 1 status ${\displaystyle Z_{1,t}}$	Player 2 status ${\displaystyle Z_{2,t}}$	Ballort ${\displaystyle B_{t}}$	comment
0			50	50	0	begin
1	5	10	45	40	−1
2	5	10	40	30th	−2
3	15th	10	25th	20th	1
4th	15th	10	10	10	2
5	10	10	0	0	2	Player 1 wins

Course of the game according to table 2

In the second example, player 1 wins by being able to hit the ball over the baseline with his last points.

t	Player 1 turn ${\displaystyle S_{1,t}}$	Player 2 turn ${\displaystyle S_{2,t}}$	Player 1 status ${\displaystyle Z_{1,t}}$	Player 2 status ${\displaystyle Z_{2,t}}$	Ballort ${\displaystyle B_{t}}$	comment
0			50	50	0	begin
1	11	3	39	47	1
2	1	10	38	37	−1
3	15th	11	23	26th	1
4th	1	9	22nd	17th	−1
5	3	6th	19th	11	−2
6th	11	3	8th	8th	1
7th	4th	3	4th	5	2
8th	1	5	3	0	−1
9	1	0	2	0	1
10	1	0	1	0	2
11	1	0	0	0	3	Player 1 wins

Game theory investigation

The attraction of the game is that choosing a high move brings the ball to the opponent's side, but at the same time leaves fewer points than the opponent for the next moves. A good "strategy" will try to keep its own positive difference small, but a negative difference rather high, in order to gain an advantage in terms of both the ball location and the remaining points.

The end phase of the game can always force victory for and one of the players (deterministic). However , there are several game outcomes for and so that the game strategy can only be aimed at increasing the probability of winning. For the analysis of the game it is important that the probability of winning depends only on the number of points of both players and the location of the ball (state of the game), but not on the number of moves that led to this state or how this state was reached ( Markov property ). ${\displaystyle S_{1,t}\leq 3}$ ${\displaystyle S_{2,t}\leq 3}$ ${\displaystyle S_{1,t}=S_{2,t}=4}$ ${\displaystyle |B_{1,t}|=1}$

Technical implementation

The game is suitable for a technical implementation in which the program learns from playing. With a starting value of 50, the state space is limited to 13005 (= 51 * 51 * 5, players can score points incl. 0 and 5 ball locations), and the game matrix is ​​the possible moves vs. the states ( ), it has about 330,000 elements, if one does not include the not allowed moves. ${\displaystyle 0\leq Z_{i}\leq S_{i}}$ ${\displaystyle S_{1},S_{2},B}$

If a move leads to a profit in the end result, it is upgraded, otherwise it is devalued. The higher the rating of a train in a given state, the greater the probability that it will be selected the next time it reaches this state. You can have such a program play against yourself, with 100,000 games already leading to strategies that achieve almost 50% probability of winning against human opponents.

variants

The game becomes more challenging if the chosen numbers are only communicated to a game master, who announces the location of the ball after the move. In this case, the point difference to the opponent - and thus also the current score of the opponent - is not known. In particular, the Markov property no longer applies to this variant, so that a technical implementation also becomes more complex.

literature

Matthias Mala: The big book of block and pencil games , Tosa Verlag, 2005, ISBN 3-85492-542-5 .