This is from Aug 27 cvs. The attached file numbers the first section with 5 rather than 1. The printed output is correct.
hawk -- Richard E. Hawkins, Asst. Prof. of Economics /"\ ASCII ribbon campaign [EMAIL PROTECTED] Smeal 178 (814) 375-4700 \ / against HTML mail These opinions will not be those of X and postings. Penn State until it pays my retainer. / \
#LyX 1.3 created this file. For more info see http://www.lyx.org/ \lyxformat 220 \textclass article \language english \inputencoding default \fontscheme default \graphics default \float_placement htn \paperfontsize default \spacing single \papersize Default \paperpackage a4 \use_geometry 1 \use_amsmath 0 \use_natbib 0 \use_numerical_citations 0 \paperorientation portrait \leftmargin 1in \rightmargin 1in \bottommargin 1in \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \quotes_times 2 \papercolumns 2 \papersides 1 \paperpagestyle default \layout Title Game Theory \layout Date Fall, MMII \layout Author Richard E. Hawkins \layout Standard Game theory can be quite useful in understanding economics. It does not seek to explain the games we play for entertainment, but to express decisions by multiple agents in the form of a game, modeling decisions \begin_inset Quotes eld \end_inset as if \begin_inset Quotes erd \end_inset individuals were players in a game. \layout Standard \begin_inset LatexCommand \tableofcontents{} \end_inset \layout Section The Prisoners' Dilemma \layout Standard Before considering games in general, let's consider the most popular game of all, the prisoners' dilemma, as seen in Figure \begin_inset LatexCommand \ref{pd1} \end_inset . This table may not have any immediate and obvious meaning, but it conveys several types of information to those who can read it. On the other hand, without a story to accompany it, it means absolutely nothing. \layout Standard \begin_inset Float figure placement htbp wide false collapsed false \layout Standard \align center \begin_inset Graphics filename pd1.ps display color rotateOrigin center \end_inset \layout Caption \begin_inset LatexCommand \label{pd1} \end_inset The basic Prisoners' Dilemma game \end_inset \layout Subsection The Story \layout Standard Larry and Moe have successfully robbed the bank while well disguised. District Attorney Curley knows this, but can't prove it--but he can prove a minor assault charge for the two of them. Larry and Moe, like all good crooks, have agreed not to rat on one another if caught, but they've been put in separate cells and can't tell whether the other is holding to the agreement. \layout Standard Curly makes an offer to the prisoners: Defect from your criminal enterprise (thus the \begin_inset Quotes eld \end_inset D \begin_inset Quotes erd \end_inset in the chart), by ratting on the other, and get a better sentence. If you confess, and the other doesn't, you can go free, but if you both confess, it's two years. If you don't confess and your partner does, it's five years. If neither confesses, they will both be locked up for six months for assault. \layout Subsection Relating the Story to the Chart \layout Standard If you look at the numbers from the sentences, you will see that in Figure \begin_inset LatexCommand \ref{pd1} \end_inset , the negative numbers are the sentence lengths in months. In each pair of numbers, the first is the \begin_inset Quotes eld \end_inset payoff \begin_inset Quotes erd \end_inset to the player on the left, or what he receives, and the second is the payoff for the player on the top. Notice that the numbers are all negative--a jail sentence is worse than nothing, and has negative value. \layout Standard To determine which pair of payoffs to use, look at each player's actions--if Larry cooperates and Moe defects, chose the first row (C) and the second column (D) for a payoff of (-60,0), meaning that since Larry kept his mouth shut while Moe ratted him out, Larry does five years while Moe walks. \layout Subsection Predicting the Outcome of the Game \layout Standard Both players must choose their moves without knowing the other's move. Nonetheless, if Larry and Moe are bright enough, we can predict what will happen in this game. Look at the game first from Larry's perspective. If Moe cooperates, keeping his mouth shut, what is the best thing for Larry to do? If Larry cooperates, he'll do six months, but he'll walk if he defects by ratting out Moe. Larry is clearly better off defecting if Moe cooperates. Now consider what happens to Larry if Moe defects. If Larry defects as well, he does two years--which increases to five if he cooperates. \layout Standard No matter what Moe does, Larry's best result comes when he defects, and Larry should defect. Moe faces the same choices, and we conclude that Moe will also defect. We predict that if both players in this game are rational, they will both defect--in spite of the fact that there is a possible outcome in which \emph on both \emph default players are better off than the outcome we reach. \layout Section Some Key Topics in Game Theory \layout Standard Having seen a simple but quite common game, we have a framework to discuss some of the problems that were involved. Let's take a closer look at these. \layout Subsection Strategy \layout Standard A player's strategy is the set of choices that a player will make at each possible point in the game, and with each possible information set (what he could know at that point) that can occur. In the simple prisoners' dilemma, there is only one such point, the beginning of the game, with no knowledge other than the payoffs of the game--the player moves blindly without knowing the other player's move, and thus the strategy is simply C or D. \layout Subsection The Payoff Matrix \layout Standard This bit of knowledge leads to the next thing that a player must have: knowledge the payoff matrix. If the player doesn't know the payoffs in the game, he cannot move in an intelligent manner. While such a game \emph on could \emph default be played, it wouldn't provide information about how people make choices. \layout Standard Keep in mind that the payoffs must represent a \emph on complete \emph default account of the payoffs in the game. In the prisoners' dilemma, the total payoff to the players actually includes much more than just the jail sentence. What happened to the money? If Larry rats out Moe, what will Moe do to Larry when he gets out? What about his conscience? The payoff should include all of these factors. \layout Subsection Normal Form Games \layout Standard This game is written in what is called \begin_inset Quotes eld \end_inset normal form. \begin_inset Quotes erd \end_inset Many games are simple enough to place into rows and columns, with each possible strategy for the first player as the label of a row, and each possible strategy for the second player labeling a column. Games with only two players and a single simultaneous move are usually explained more clearly in this form. The extended form will be discussed later. \layout Subsection Non-Cooperative Games \layout Standard The prisoners' dilemma is a \begin_inset Quotes eld \end_inset non-cooperative \begin_inset Quotes erd \end_inset game. This means that the players cannot make binding agreements with each other in the game. The players would be better off if they could make an agreement to cooperate, seeing six month sentences rather than two year sentences. In a cooperative game, such agreements are allowed. \layout Section The Terrorist Game \layout Standard Not all games have players moving simultaneously in ignorance of the moves of the other players. In the terrorist game, the terrorist has taken control of the plane and has a bomb. He demands a million dollars in return for not blowing up the plane. This game is shown in Figure \begin_inset LatexCommand \ref{tr1} \end_inset . \begin_inset Float figure placement htbp wide false collapsed false \layout Standard \align center \begin_inset Graphics filename tr1.ps display color rotateOrigin center \end_inset \layout Caption \begin_inset LatexCommand \label{tr1} \end_inset The Terrorist Game \end_inset The first move is for the airline, which can either pay (p) or not pay (n). The \begin_inset Quotes eld \end_inset 1 \begin_inset Quotes erd \end_inset at the top of the diagram is to show that it is player 1, the airline, that moves at this point. Each decision has a different \begin_inset Quotes eld \end_inset node, \begin_inset Quotes erd \end_inset indicated by the two black dots separated by a dotted line. In this form, player 2, the terrorist, \emph on does know \emph default which move player 1 has chosen. The information sets at the two nodes are therefore different--each contains different information about how player 1 moved. The second move is for the terrorist, who can explode the bomb (B) or not explode it and live (L). If the terrorist explodes the bomb, it costs the airline an additional five million dollars, and the terrorist dies, for a payoff of -10. \layout Standard In deciding how to move, the airline should consider what the terrorist \emph on would \emph default do in each possible situation. If the money has been paid, the terrorist presumably has no reason to blow up the plane, takes the million for a payoff of 1, while the airline has a payoff of -1 from paying the ransom. On the other hand, if the airline does not pay, consider the terrorists choices: die with a payoff of -10, or live with a payoff of -1 for going to prison. We therefore predict that the airline will not believe that the terrorist will blow up the plane, and will refuse to pay, followed by the terrorist choosing to live rather than carrying out the threat. \layout Section New Concepts from the Terrorist Game \layout Subsection Extensive Form Games \layout Standard The terrorist game has been shown in \begin_inset Quotes eld \end_inset extensive form, \begin_inset Quotes erd \end_inset in which the possible paths the game can follow are drawn out. Much longer games are possible, and will be considered later. \layout Subsection Information Sets \layout Standard In this game, the terrorist could arrive at two different information sets, depending upon the choice of the airline. The information set for the terrorist in this case includes the move of the prior case, but this is not always the case. Suppose the airline is given a third choice, lying (l) that the ransom has been paid, as in Figure \begin_inset LatexCommand \ref{tr2} \end_inset . \begin_inset Float figure placement htbp wide false collapsed false \layout Standard \begin_inset Graphics filename tr2.ps display color rotateOrigin center \end_inset \layout Caption \begin_inset LatexCommand \label{tr2} \end_inset Terrorist Game where the airline can lie \end_inset The terrorist has the same information set whether the airline pays or lies, even though there are two decision nodes within the set. The combined set is noted by the ellipse around the nodes. \layout Standard The prisoners' dilemma can also be written in extensive node as in Figure \begin_inset LatexCommand \ref{pd2} \end_inset . \begin_inset Float figure placement htbp wide false collapsed false \layout Standard \begin_inset Graphics filename pd2.ps display color rotateOrigin center \end_inset \layout Caption Prisoners' Dilemma in extensive form \begin_inset LatexCommand \label{pd2} \end_inset \end_inset While the game was described as simultaneous, what if the prisoner^, k, make their decision in separate rooms a minute apart? In this case, the extensive form may more accurately describe the situation faced by the second prisoner. \layout Subsection Credible Threats \layout Standard Perhaps the most important notion here is that of a credible threat--which the terrorist's is not. If the terrorist actually \emph on would \emph default blow up the plane if not paid, the airline would pay. But the threat simply isn't credible, as the terrorist dies as well. Later we will discuss the possibility of the terrorist somehow irrevocably committing himself to blowing up the plane in this circumstance \emph on before \emph default the airline's move, and how this would change the game. We will also discuss experiments which show that people sometimes \emph on will \emph default take a loss in such a situation, even though we find it irrational. \the_end