Game Theoryand Business Strategy Part I 陳坤銘 政大國貿系
Game Theoryand Business Strategy Part I Lecture 4 Games with Sequential Moves
Experiment 4:Single-Offer Bargaining (The Ultimatum Game) • Two players A and B are chosen. Player A offers a split of 10 dollars. If B agrees, both get paid the agreed coins and the game is over. • If B refuses, it is B’s turn, but now the sum is only 8 dollars. If A accepts B’s offer, the two get paid the agreed coins. If A refuses, the game is over and neither gets anything.
Experiment 5: Adding Numbers (Win at 100) • Two players take turns choosing a number between 1 and 10 (inclusive). • A cumulative total of their choices is kept. The player to take the total exactly to 100 is the winner.
Outline • Game trees • Solving games by using tree • Adding more players • Order advantage • Adding more moves • Evidence concerning rollback • Strategies in the survivor game
( 2, 7, 4, 1) Up ANN ( 1, -2, 3, 0) Down 1 BOB High ( 1.3, 2, -11, 3) 2 DEB Stop Low ( 0, -2.718, 0, 0) 3 ANN ( 10, 7, 1, 1) Good 50% Risky Go ( 6, 3, 4, 0) Nature CHRIS Bad 50% ( 2, 8, -1, 2) Safe ( 3, 5, 3, 1) GameTree (Extensive Form of a Game) PAYOFFS a Four players: ANN, BOB, CHRIS, DEB Branches Terminal Nodes Initial Node Figure 3.1 An illustrative Games Tree
Node • Decision (action) node: the player chooses an action at that node. • Initial node (root): the starting point of game • Terminal node: the last node along each path.
Branches • Represent the possible actions that can be taken from any decision node. • There must be at least one branch leading from each decision node. • Every decision node can have only one branch leading to it.
Uncertainty and Nature’s Moves • Something happens at random. • Use of the player Nature allows us to introduce uncertainty in a game and gives us a mechanism to allow things to happen that are outside the control of any of the actual players.
Strategy • A complete plan of action, e.g., we might state one of Ann’s strategies in Fig. 3.1 as:“ Choose Go at the first move, and choose Up if the next move arises”. • Specification of strategies has to do with stability of equilibrium., when players’ choices were subject to small disturbances.
The Smoking Game: -1 Continue Try Not Carmen 1 Not 0 Figure 3.2 The Smoking Decision
-1,1 Continue Future Carmen Try Not 1,-1 Today’s Carmen Not 0 Solving Games by Using Trees Figure 3.3 The Smoking Game
Solving Games by Using Trees (con’t) • We cut off, or prune, the branches “NOT” emerging from the second move, since at that node the action associated with that braches will not been chosen.. • We then continue to prune one of the branches “NOT” emerging from the first node.
Solving Games by Using Trees (con’t) • The “fully pruned” tree leaves only one branch emerging from the initial node and leading to a terminal node. • Following the only remaining path through the tree shows what might happen in the game when all players make their best choices with correct forecasting of all future consequences. • Alternative way of showing player choices is to “highlight” the branches that are chosen.
-1,1 Continue Future Carmen Try Not 1,-1 Today’s Carmen Not 0 The Smoking Game Figure 3.4 Pruning the Tree of the Smoking Game
-1,1 Continue Future Carmen Try Not 1,-1 Today’s Carmen Not 0 The Smoking Game Figure 3.5 Showing Branch Selection on the Tree of the Smoking Game
Rollback • A method that looks ahead and reasons back to determine behavior in the sequential-move game. • Also called Backward Induction. • Require starting to think about what will happen at all the terminal nodes and literally “rolling back” through the tree to the initial node when doing your analysis.
Equilibrium • When all players are rational calculators in pursuit of their respective best payoffs. • When all players choose their optimal strategies found by doing rollback analysis, this set of strategies is called the rollback equilibrium of the game.
3, 3, 3 Contribute Talia Contribute 3, 3, 4 Don’t Nina 3, 4, 3 Contribute Contribute Don’t Talia Don’t 1, 2, 2 Emily Contribute 4, 3, 3 Talia Contribute Don’t 2, 1, 2 Don’t Nina Contribute 2, 2, 1 Don’t Talia Don’t 2, 2, 2 Adding More Players Equilibrium path of play Figure 3.6 The Street Garden Game
Equilibrium path of play • Miss most of the branches on nodes • Choices made early in the game are affected by players’ expectations of what would happen if they chose to do something other than what was best for her (out-of-equilibrium).
How to find the equilibrium • List the available strategies for each player. • Find the optimal strategy, or complete plan of action for each player. • Find the actual path of play in the rollback equilibrium, found by putting together the optimal strategies for all the players.
Available Strategies • In Fig. 3.6, available strategies for Emily: (C, D) • Available strategies for Nina: (CC, CD, DC, DD) • Available strategies for Taila: (CCCC,CCCD,CCDC,CCDD, CDCC,CDCD,CDDC,CDDD, DCCC,DCCD,DCDC,DCDD, DDCC,DDCD,DDDC,DDDD) 4. Rollback eq. of the Game: D for Emily, DC for Nina and (DCCD) for Taila.
FDE, Congress Blance 4, 3 Congress Low 1, 4 Deficit FED Balance 3, 1 High Congress Deficit 2, 2 Changing the order of moveCase 4: Both players may do better Sequential play: FED moves first Simultaneous play Sequential play: Congress moves first Congress, FED 3,4 FED Low Balance 1,3 High Congress Low 4, 1 Deficit FED High 2, 2
Showing and analyzing sequential-move games in strategic form SPE: the strategy is optimal in every subgame for the player whose turn it is to act at that node, whether or not the node and the subgame lie on the equilibrium path of play. Congress, FED 3,4 FED Low Two Nash equilibrium Only one Subgame-perfect equilibrium (SPE) Balance 1,3 High Congress Low 4, 1 Deficit FED High 2, 2
Sequential Rationality COMMANDMENT Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today
Order advantages • First-mover advantage: comes from the ability to commit oneself to an advantageous position and to force the other players to adapt it. • Second-mover advantage: comes from the flexibility to adapt oneself to the others’ choices.
Adding more moves In a tic-tac-toe game, two player (X and O) each try to be the first to get two of their symbols to fill any row, column, or diagonal of a two by two game board. There are 24 terminal nodes to consider (refer to fig. 3.7). The first player always win.
Adding more moves (con’t) • Tic-Tac-Toe and Chess games are sometimes truly complex. • A game may be amenable in principle to a complete solution by rollback, its complete tree may be too complicated to permit such solution in practice.
Adding more moves (con’t) • We must use a combination of two methods to solve a complicated game: • Calculation based on the logic of rollback; • Rules of thumb for valuing intermediate positions on the basis of experience (intermediate valuation function).
Accept A B x (x, 100-x) Reject (0,0) Evidence concerning rollback Player A proposes a split to share 100 dollars between A & B. If B accepts this proposal, the 100 dollars are divided as proposed by A. If B rejects the proposal, neither player gets anything. • Rollback predicts that B should accept any sum, no matter how small. A should propose “x=99” (99 to me, 1 to B). • This particular outcome almostnever happens. 50-50 is the single most common proposal.
The Centipede Game • Two players A and B are chosen. The experimenter puts dime on the table. It A take it’s the dime, the game is over, with A getting the 10 cents and B getting nothing. • If A passes, the experimenter adds a dime, and now B has the choice of taking the 20 cents or passing. • The turns alternate, and the pile of money grows until reaching some limit—say, a dollar—that is know in advance by both players.
A B A B A B Pass Pass Pass Pass Pass Pass (0, 0) Take dime Take dime Take dime Take dime Take dime Take dime (10,0) (0,20) (30,0) (0,40) (90,0) (0,100) The Centipede Game (cont.) • A should take the very first dime and end the game. • In most experimental settings, such games go on for at least a few rounds.
Implications of These Experiments • The apparent violations of strategic logic can be explained by recognizing that people do not care merely about their own money payoffs, but internalizing concepts such as fairness. • However, not all observed plays, contrary to the precepts of rollback, have some such explanations. People do fail to look ahead far enough, and they do fail to draw the appropriate conclusions from attempts to look ahead. • Nevertheless, the game-theoretical analysis of rollback and rollback equilibrium serves an advisory or prescriptive role as much as it does a descriptive role.
Strategies in the survivor game • The strategic problems facing all players: • To be generally regarded as a productive contributor to the tribe’s search for food and other tasks of survival, but to do so without being regarded as too strong a competitor and therefore a target for elimination; • To form alliances to secure blocks of votes to protect oneself from being voted off; • To betray these alliances when the numbers got too small and one had to vote against someone; • To betray alliances without seriously losing popularity with the other players, who would ultimately have the power of the vote on the jury.
Rich Prob. 0.4 Rich wins Prob.= 0.18 Jury picks Kelly Kelly Prob. 0.6 Rich keeps Kelly wins Rich Prob. 0.45 Jury picks Ruby Ruby Ruby wins Rich Prob. 0.6 Rich wins Prob.=0.3 Jury picks Nature Kelly Prob. 0.5 Rich Kelly Prob. 0.4 Kelly wins Kelly keeps Jury picks Continue Ruby Ruby Prob. 0.05 Ruby wins Ruby Ruby keeps Jury picks Rich Ruby Ruby wins Rich Rich Prob.=0.54 Rich Prob. 0.6 Jury picks Give up Rich Kelly Prob. 0.4 Kelly wins Kelly keeps Kelly Prob. 0.9 Jury picks Ruby Ruby Ruby wins Nature Ruby Prob. 0.1 Ruby keeps Jury picks Rich Ruby Ruby wins Survivor Immunity Game Tree