Adversarial Search and Game Theory

Abstract

Game theory is the formal study of conflict and cooperation, first time introduced as long back as 1921 by mathematician Emile Borel, then enriched in 1928 by von Neumann and Oskar Morgenstern, and much enriched by Josh Nash, has enormous applications, including    in business, and even in the prediction of election results, etc. The game playing is also a search process. The chapter presents the classes of games as combinatorial and games of chance, then further as zero-sum games and non-zero-sum games, the prisoner’s dilemma, game playing strategies, the games of perfect information, arbitration scheme in games, minimax search in game playing, and analysis of specific games like tic-tac-toe. The more efficient search processes like alpha and beta are presented, as well as the alpha cutoff and beta cutoff methods to prune the search process are presented and analyzed, followed with chapter summary, and an exhaustive list of exercises along with a number of multiple-choice questions provided at the end.

Exercises

• 1. Demonstrate the min-max search for the tic-tac-toe puzzle (for 10 moves).

• 2. Demonstrate the alpha-beta search for the tic-tac-toe puzzle.

• 3. Apply alpha-beta search (from left to right) to the game-tree in Fig. 11.18. Show the backed-up value of each node. Mark with an X any branches that are not searched. Identify these as alpha/beta cutoffs. Mark the best move with an arrow from the root node.

  • a. Identify these as alpha/beta cutoffs, mark the best moves with an arrow from the root node.

  • b. Compute the time complexity of search with worst-case branching factor of 3 and height of the tree as h.

• 4. A game nim is played as follows: there are two players who remove one, two, or three coins, alternately from a stack of five coins. Represent this game playing as a search tree. Suggest any suitable strategy for winning this game?

• 5. Given the tree in Fig. 11.19, explore this tree using the alpha-beta procedure. Indicate all parts of this tree that are cutoff. Also, indicate the winning path(s), and strike out all the values that are not required to be computed.

  • a. Identify these as alpha/beta cutoffs, mark the best moves with an arrow from root node.

  • b. compute the time complexity of search with worst-case branching factor of 3 and height of tree as h.

• 6. Consider the min-max tree shown in the Fig. 11.20, whose leaves are labeled with natural numbers; n and m are variables.

  • a. Assign values to n and m such that, to compute the value at the root node, no alpha-beta cutoff is possible. Compute the value of the root node.

  • b. Assign values to n and m such that, to compute the value at the root node, an alpha-beta cutoff is possible. Indicate the cut and compute the value of the root node.

• 7. Following are either two-player or many player games, and standard minimax/alpha-beta search techniques are to be applied for game playing. Explain, how well these techniques apply to these games? Give the formal approach for each, where possible.

  • a. Basketball

  • b. Badminton

  • c. Football

  • d. Soccer

  • e. Tennis

  • f. Kabbadi

• 8. Consider two firms A and B, that give service in the same market. The firms incur constant average costs of $2 per unit, and are free to choose a high price of $10 or a lower price of $5 per unit for marketing. When both firms set a high price, the total demand is 5,000 units which are split evenly between the two firms. When both set a low price, the total demand is 10,000 units, which is again split evenly. If one firm sets a low price and the second a high price, the low priced firm sells 8,000 units, and the high priced firm only 1,000 units. Analyze the pricing decisions of the firms A and B as a noncooperative game.

  • a. Construct the payoff matrix, where the elements of each cell of the matrix are the two firms’ profits.

  • b. Derive the equilibrium set of strategies.

  • c. Is it the game of prisoners’ dilemma? Justify.

• 9. Find out the Nash equilibrium for the game shown in Fig. 11.21, which is played between two-players A and B, having strategies $s_1, s_2, s_3$ and $s_1^\prime , s_2^\prime , s_3^\prime$, respectively.

• 10. Consider the Table 11.1 for two-player game.

  • a. Find out the maximum moves by the game to win/loose.

  • b. Which moves are dominated?

  • c. Identify all the best responses.

  • d. Is there any Nash equilibrium?

• 11. Draw a payoff matrix for the following game of scheduling a party. Also, find maximin moves, domination, best responses, and Nash equilibrium (if it exists). Let the friends A and B do not speak to each other, but have many common friends. Both of them want to invite these friends to the party, either on Saturday or Sunday. However, both of them prefer Sunday over Saturday. If both decide the party for the same day, it will be considered as a disaster with a (negative) payoff of $500 for both. If they plan the party on different days, the one proposing Sunday gets a payoff of $250, and the other of $200.

• 12. For two-players game shown in Table 11.2, find out all Nash equilibria states, possible domination, maximin moves, and best responses.

• 13. Find out the saddle point for the game shown in Fig. 11.5.

• 14. What can be the true considerations, for example, to reach to Nash equilibrium faster in a buyer vs seller game?

• 15. Are we in a position to determine in advance that arbitration between two parties will result in an equilibrium state? Justify for yes/no.

• 16. Justify the following statement: “It is often found that Nash arbitration is often unfair, in which rich becomes richer and poor becomes poorer.”

• 17. For any minimax game of even height game-tree, show that final results of the game shall be the same, irrespective of who plays first.

• 18. Apply the minimax algorithm for the problem of prisoner’s dilemma (Fig. 11.2), and demonstrate the backup of values from static levels.

• 19. Why the game of tic-tac-toe is BFS search? Justify.

• 20. Write an algorithm, to compute the weight of any general configuration of tic-tac-toe given any number of Xs and Os on arbitrary position. Note that if $0 \le |X| \le 5$ then $0 \le |O| \le 4$ and vice versa.

• 21. In a game search tree with the root node at level 0, suppose the alpha cutoff occurs at level i. What is the minimum required depth of the tree? Justify your answer.

• 22. For a game-tree of depth d, with branching factor b uniform at all the sub-tree, let alpha cutoff occurs at every alternate depth due to the second node from left in each sub-tree. What is an expression for the time-complexity of the search using only alpha cutoff? Assume that the time-complexity of the minimax algorithm is $O(b^d)$.

• 23. Assume that in a game-tree, the alpha and beta cutoff occurs at every alternate level. Find out the time complexity of this search.

• 24. Show that if static nodes are ordered in order of their static values, then the time complexity of alpha-beta search is $O(b^{\frac{d}{2}})$.

• 25. Show that if static node weights are in random order, then the time complexity of alpha-beta search is $O(b^{\frac{3d}{4}})$.

• 26. In the Fig. 11.15, find out the alpha-beta cutoffs in each case, when order of sub-trees are $P_3P_2P_1$, $P_2P_3P_1$, and $P_2P_1P_3$. What you conclude by this change in order?

• 27. Are the winning results of minimax and alpha-beta search tree identical for a given search tree? Justify.

• 28. Show that cost functions of symmetric configurations in tic-tac-toe are identical.

• 29. Two firms Alpha and Beta serve the same market. They have constant average costs of $2 per unit. The firms can choose either a high price ($10) or a low price ($5) per unit for their product. When both firms set a high price, the total demand is 10,000 units, which is split evenly between the two firms. When both set a low price, the total demand is 18,000 units, which is again split evenly. If one firm sets a low price and the other a high price, the low priced firm sells 15,000 units, while the high priced firm sells only 2,000 units. Analyze the pricing decisions of the two firms as a noncooperative game.

  • a. In the scenarios mentioned above, form representation, construct the payoff matrix, where the elements of each cell of the matrix are the two firms’ profits.

  • b. Derive the equilibrium set of strategies.

  • c. Explain why this problem an example of the prisoners’ dilemma game.

• 30. Write a parallel search algorithm for an alpha-beta search to reduce the search space.

• 31. Select one or more than one, or True/False, or nil answer in the following multiple-choice questions:

  • a. Prisoner’s dilemma is following type(s) of game:

    (A) Noncooperative game        (B) Non-zero-sum game

    (C) Imperfect information game (D) all the above

  • b. Poker is following type(s) of game:

    (A) Zero-sum        (B) Perfect Information

    (C) Non-zero sum (C) None of above

  • c. Which of the following are static games?

    (A) Prisoner’s dilemma (B) Bid of winning a contract

    (C) Chess        (D) Poker

  • d. Which of the following are called simultaneous move games?

    (A) Prisoner’s dilemma (B) Bid of winning a contract

    (C) Chess        (D) Poker

  • e. The tic-tac-toe is following type of search:

    (A) Iterative deepening (B) DFS

    (C) Best-first search        (D) BFS

  • f. Alpha is defined by the values of successors of:

    (A) MAX position        (B) MIN position

    (C) Both MAX and MIN (D) None of above

  • g. Alpha cutoff occurs among the successors of:

    (A) MIN position        (B) MAX position

    (C) Depends of the structure of the tree (D) None of above

  • h. Beta is defined by the values of successors of:

    (A) MAX position        (B) MIN position

    (C) Can be any position (D) None of above

  • i. Beta cutoff occurs among the successors of:

    (A) MIN position        (B) MAX position

    (C) Depends of the structure of the tree (D) None of above

  • j. The alpha-beta search belong to the following class of algorithm:

    (A) Hill-climbing       (B) Branch-and-bound

    (C) Divide-and-conquer (D) Iterative approximation

  • k. Alpha is defined at odd level of the tree (T/F)?

  • l. Beta is defined at even level of the tree (T/F)?