Tennis (paper game)
Tennis is a strategic paper - and-pencil game for two players.
Regulate
The playing field consists of four fields and a center line. These are referred to as (-2, -1,0,1,2 ), wherein the negative fields player 1 and the positive belong to Player 2. At the start when the ball is in "0". Starting from a starting point number (eg 50 ) for both players, each player chooses a number, and the ball is moved in the direction of the player with the smaller number. The selected number reduces the score of the player. The aim of the game is to knock the ball over the baseline of the other player.
Mathematical Description
For the simple mathematical representation of the ejection is taken above the baseline in the field name. The playing field is thus in -3, -2, -1,0,1,2,3 divided, with exit -3 and 3 the game.
The train of player i at time t is denoted by, the ball is at time t in.
Applies at the beginning and is for both players.
Each player chooses an integer, which, may be selected for. The selected number reduces the player () the points.
Then, the following rules apply for the ball:
- If, then the ball is moved in the direction of player 2. If the ball was on the side of player 1, he is beaten back over the center line ().
- Otherwise, the ball is moved one space ().
For the outcome of the game is determined:
- The game is finished when the ball is hit on one of the basic lines either, or if no player has more points.
- In both cases, the game for the player is deemed lost, on whose side is the ball.
- ( Until a certain point limit reached) If you play more than once, you can specify that for the ejection of the baseline gain 2 points are awarded, otherwise only 1 point win.
Example Games
In the first example, Player 1 wins, after both players have no more points ( ball is still in the field).
In the second example, Player 1 wins by being able to hit the ball on the baseline out with his last points.
Game Theoretical investigation
The appeal of the game is that the choice of a high train while bringing the ball on the opponent's side, but at the same time remain fewer points than the opponent for the upcoming trains. A good "strategy" will try to maintain their own positive difference is small, but a negative difference tends to be high in order to get advantage in both the Ballort as well as the remaining points.
The final stage of the game can be used for and one of the players always win force ( deterministic). Even for and there are several endings, so that the game strategy can be aligned only to the increase in the probability of winning. For the analysis of the game is important that the probability of winning depending only on the number of points of both players and Ballort ( state of game), but not the lead to the number of trains on this condition or is in what way this condition is reached ( Markov property).
Technical implementation
The game is suitable for programmatic implementation, in which the program learns from the games. The state space is at a start value from 50 to 13005 ( = 51 * 51 * 5, point of each player including 0 and 5 ball locations) limited and the game matrix are the possible moves vs. the states (), it has about 330000 members when they do not receive the non- legal moves.
When a train in the final to win leads, he is upgraded, otherwise devalued. The higher the rating of a train is at a given state, the greater will be the likelihood that it is selected the next time this state is reached. You can let them play such a program against itself, which produces 100,000 games lead to strategies that achieve against human opponents almost 50 % winning percentage.
Variants
The game gets more challenging if the selected numbers are notified only a game manager who announces the Ballort after the train. In this case, the point difference to the opponent - and thus also the current score of the opponent - not known. In particular, for this variant no longer the Markov property, so that even a technical program implementation is more complex.