Fundamentals of
Game Theory

Author

Affiliation

Paulo Fagandini

Prague University of Economics and Business
Faculty of Informatics and Statistics

Introduction

Core Bibliography

The core of the contents of this chapter takes as reference the book

:book: Strategies and Games: Theory and Practice by Prajit K. Dutta. 2001 MIT Press.

Strategic Form Games

We will focus the utilization of Strategic Form games to model simoultaneous games, and therefore the timing of the game becomes unnecessary. We will just need the set of players, their strategies, and the payoffs associated to any strategy profile.

Strategic Form Games

Let’s name our players \(P1\) and \(P2\)
Let \(S_i\) be the set of strategies available for player \(i\)
Say \(\pi_i(s_1,s_2)\) be the payoff for player \(i\), with \(s_1\in S_1\) and \(s_2\in S2\).

Strategic Form Games

Suppose \(S_1=\{U,D\}\) and \(S_2=\{L,R\}\), the payoff matrix would be:

1\(\setminus\) 2	\(L\)	\(R\)
\(U\)	\((\pi_1(U,L),\pi_2(U,L))\)	\((\pi_1(U,R),\pi_2(U,R))\)
\(D\)	\((\pi_1(D,L),\pi_2(D,L))\)	\((\pi_1(D,R),\pi_2(D,R))\)

. . .

Important

The payoff for each player, depends on the strategy profile, that in each position on the table, is the same.

Strategic Form Games

What if we have more than 2 players, and strategies?

. . .

If we have \(N\in\mathbb{N}\) players, we will index them: 1,2, \(\dots\), \(N\).
Their strategy space is \(S_i\).
A strategy profile is an ordered tuple \((s_1,s_2,\dots,s_N)\in S_1\times S_2\times\dots S_N\).
\(\pi_i(s_1,\dots,s_N)\) represents the payoff for player \(i\), given the strategy profile \((s_1,s_2,\dots,s_N)\).

. . .

We can do better

Strategic Form Games

Given that the strategies of the other players (other than player \(i\)) are out player’s \(i\) control, we will summarize them as \(s_{-i}\). So we will write:

\[\pi_i(s_1,\dots,s_N) = \pi_i(s_i,s_{-i}) \]

And \(s_{-i}\) represents all chosen \(s_j\) when \(j\neq i\). We could say the other’s strategies in the current strategy profile.

The Prissoner’s Dilemma

This is a classical game, and this is a great opportunity to introduce it:

Two prissoners, Calvin and Klein, are hauled in for a suspected crime. The \(DA\) speaks to each prisoner separately, and tells them that she more or less has the evidence to convict them but they could make her work a little easier (and help themselves) if they confess to the crime. She offers each of them the following deal: “Confess to the crime, turn a witness for the State, and implicate the ohter guy, and you will do no time. Of course, your confession will be worth a lot less if the other guy confesses as well. I that case, you both go in for 5 years. If you do not confess, however, be aware that we will nail you with the other guy’s confession, and then you will do fifteen years. In the event that I cannot get a confession from either of you, I have enough evidence to put you both away for a year.”

Prajit K. Dutta’s version of this game, in the book Strategies and Games: Theory and Practice. MIT Press, 2001.

The Prissoner’s Dilemma

Note that this would be something like a reverse payoff matrix, as as a payoff we are writing down the years in prison, and most likely, the lower the number of years in prison, the better for both prissoners!

Calvin - Klein	Confess	Not Confess
Confess	\((5,5)\)	\((0,15)\)
Not Confess	\((15,0)\)	\((1,1)\)

We can turn this table into something more traditional, assuming that each year in prison, equals a payoff of -1.

The Prissoner’s Dilemma

Calvin - Klein	Confess	Not Confess
Confess	\((-5,-5)\)	\((0,-15)\)
Not Confess	\((-15,0)\)	\((-1,-1)\)

Suppose Calvin thinks Klein will confess. In that case, Calvin would be much better confessing! (\(-5 > -15\)).
Suppose Calvin thinks Klein will Not Confess. In that case, Calvin would also be much better off by confessing! (\(0 > -1\))

The Prissoner’s Dilemma

Symmetrically, these options are the same for Klein. Note that confess is a dominant strategy, because independently of what the other decides, confess always generates at least as much payoff as any alternative. You cannot improve over the payoff you can get with confess, no matter what your counterpart is doing.

Equivalence in the Extensive form

We said, these are simoultaneous games, but we do not need that both prissoner’s actually chose simoultaneously. It is enough that when one decide, the other does not know what the first prisoner’s chose.

Equivalence in the Extensive form

Think on the Rock-Paper-Scissors game. As both players can see each others hands, simultaneity is necessary.

Suppose now that we introduce a referee. One player tells secretely to the referee what her choice is (rock, paper, or scissors). Then the referee asks the other player, separately what is her own choice. Finally the referee can reveal if there is a winner or a draw. The game works just fine, even though one player played first.

Equivalence in the Extensive form

The key was that the player that played last, did not know what the other player played when she made her decision, and also that the first player, knew that they other player was unaware of her move by the time she made her decision.

Equivalence in the Extensive form

This is the Strategic or Normal form to represent this game:

1-2	\(R\)	\(P\)	\(S\)
\(R\)	\((0,0)\)	\((-1,1)\)	\((1,-1)\)
\(P\)	\((1,-1)\)	\((0,0)\)	\((-1,1)\)
\(S\)	\((-1,1)\)	\((1,-1)\)	\((0,0)\)

Equivalence in the Extensive form


\usetikzlibrary{calc}

\tikzset{
  solid node/.style={circle,draw,inner sep=1.5,fill=black},
  hollow node/.style={circle,draw,inner sep=1.5}
}

\begin{tikzpicture}[scale=1.5,font=\footnotesize]
  \tikzstyle{level 1}=[level distance=15mm,sibling distance=35mm]
  \tikzstyle{level 2}=[level distance=15mm,sibling distance=10mm]

  \node(0)[solid node,label=above:{$P1$}]{}
    child{
      node(1)[solid node, label=above left:{$P2$}]{}
      child{node[hollow node,label=below:{$(0,0)$}]{} edge from parent node[left]{$R$}}
      child{node[hollow node,label=below:{$(-1,1)$}]{} edge from parent node[right]{$P$}}
      child{node[hollow node,label=below:{$(1,-1)$}]{} edge from parent node[right]{$S$}}
      edge from parent node[left,xshift=-3]{$R$}
    }
    child{
      node(2)[solid node, label=above right:{$P2$}]{}
      child{node[hollow node,label=below:{$(1,-1)$}]{} edge from parent node[left]{$R$}}
      child{node[hollow node,label=below:{$(0,0)$}]{} edge from parent node[right]{$P$}}
      child{node[hollow node,label=below:{$(-1,1)$}]{} edge from parent node[right]{$S$}}
      edge from parent node[right,xshift=3]{$P$}
    }
    child{
      node(3)[solid node, label=above right:{$P2$}]{}
      child{node[hollow node,label=below:{$(-1,1)$}]{} edge from parent node[left]{$R$}}
      child{node[hollow node,label=below:{$(1,-1)$}]{} edge from parent node[right]{$P$}}
      child{node[hollow node,label=below:{$(0,0)$}]{} edge from parent node[right]{$S$}}
      edge from parent node[right,xshift=3]{$S$}
    };

   \draw[dashed,rounded corners=10]($(1) + (-.2,.25)$)rectangle($(3) +(.2,-.25)$);
   \node at ($(1)!.5!(2)$) {$P2$};

\end{tikzpicture}

The dashed set represents an information set. This means that all these nodes carry the same information, or that at this point \(P2\) does not know exactly in which node she is, and must play accordingly. This is equivalent to a simoultaneous game.

Dominant Strategy Solution

Remember the dominan strategy in the Prissoner’s dilemma?

Confess was dominant, because no matter what the other player did, confessing always yielded the highest payoff.

Dominant Strategy Solution

Strongly Dominant Strategy

A strategy \(s_i'\) is said to be strongly dominant if \[\pi_i(s_i',s_{-i}) > \pi_i(s_i,s_{-i})\quad \forall s_i\in S_i,\ \forall s_{-i}\in\prod_{j\neq i} S_j\]

Dominant Strategy Solution

Weakly Dominant Strategy

A strategy \(s_i'\) is said to be weakly dominant if \[\pi_i(s_i',s_{-i})\geq \pi_i(s_i,s_{-i})\quad \forall s_i\in S_i,\ \forall s_{-i}\in\prod_{j\neq i} S_j\] and \[\exists \hat{s}_{-i}\in\prod_{j\neq i} S_j\ s.t.\ \forall s_i \in S_i\setminus\{s_i'\},\ \pi_i(s_i',\hat{s}_{-i})>\pi_i(s_i,\hat{s}_{-i})\]

Application: Second Price (Sealed Bid) Auctions

Suppose there is an auction, for which \(N\) bidders are competing. Each bidder \(i\) assigns a value \(v_i\) to the object being sold. The auction works like this:

Every bidder submit a bid \(b_i\) in a sealed envelope.
Once all bids are collected, the auctioneer opens the envelope, and announces the winner.
The winner pays the value of the second highest bid, and gets the object. All other bidders walk out without the item, but without paying anything.

Application: Second Price Auctions

Any single bidder faces the following dilemma:

If I bid too much, I will win for sure, but I might end up overpaying.
If I bid too little, I will lose the auction and walk home empty handed.

Application: Second Price Auctions

If the bidder gets the item, the payoff is \[\pi_i(b_i,b_{-i})=v_i-\max\{b_j\}_{j\neq i}\]
If the bidder loses, the payoff is \[\pi_i(b_i,b_{-j})=0\]
The expected payoff is then \[E\left[\pi_i(b_i,b_{-i})\right]=Pr\left(b_i>b_{-i}\right)\left(v_i-\max\{b_j\}_{j\neq i}\right)\]

Application: Second Price Auctions

The question is then, what is the value of \(b_i\) that maximizes the expected payoff? i.e. what is \(b_i^*\)?

Suppose we chose \(b_i^*>v_i\).
Suppose we choose \(b_i^*<v_i\).
What about then if we choose \(b_i^*=v_i\)?

. . .

This is the solution proposed by William Vickrey, and these types of auctions are known as Vickrey Auctions.

Application: Second Price Auctions

These auctions present the bidder with some tradeoffs:

Enforces true revelation (bidders reveal their true valuations).
It is very simple and transparent.
Might, under some reasonable assumptions, generate lower prices than other auction formats.
Bidders might skip these auctions if they have reasons to hide their valuations from competitors.

Application: Second Price Auctions

Why now?

. . .

This auction presents an example of a dominant strategy, we saw that \(b_i=v_i\) always provides a higher (or equal) than any other bid, and therefore it is a weakly dominant strategy, whatever the other bidders are doing.

Dominance Solvability

Dominated Strategy

A strategy \(s_i^\#\in S_i\) is dominated by another strategy \(s_i'\in S_i\), if the latter does at least as well as \(s_i^\#\) against every strategy of the other players, and against some it does strictly better, such that:

\[\pi_i(s_i',s_{-i})\geq\pi_i(s_i^\#,s_{-i}),\quad \forall s_{-i}\in\prod_{j\neq i}S_j\] and \[\exists \hat{s}_{-i}\in\prod_{j\neq i} S_j,\ s.t.\ \pi_i(s_i',\hat{s}_{-i})>\pi_i(s_i^\#,\hat{s}_{-i})\]

If a strategy is not dominated, then it is said to be undominated.

Dominance Solvability

Corollary

If a strategy is dominated, it cannot be optimal.

Undominated vs Dominant strategy

Of course, looking at the definition, a dominant strategy is optimal, however, not every game has a dominant strategy for every player!

Dominance Solvability

How can we solve this problem?

Iterated Elimination of Dominated Strategies

A first approach to start looking for a solution, could be to reduce a game by removing dominated strategies, i.e. all the strategies for which there is always a better alternative whatever the other players chose to do.

Iterated Elimination of Dominated Strategies

Consider the following game

P1 - P2	Left	Right
Up	(1,1)	(0,1)
Middle	(0,2)	(1,0)
Down	(0,-1)	(0,0)

For \(P1\) there is no dominant strategy, but there is a dominated strategy, Down.

Iterated Elimination of Dominated Strategies

Consider the following game

P1 - P2	Left	Right
Up	(1,1)	(0,1)
Middle	(0,2)	(1,0)

For \(P2\) now, Right provides payoffs that are lower or equal than Left, and therefore it is dominated.

Iterated Elimination of Dominated Strategies

Consider the following game

P1 - P2	Left
Up	(1,1)
Middle	(0,2)

For \(P1\) now, Up is clearly the best choice, and the solution is therefore \((Up,Left)\). In this case it is said to be reached by iterated elimination of dominated strategies (IEDS). The game itself is said to be dominance solvable.

Bibliography

These slides took definitions and examples from the book:

Strategies and Games: Theory and Applications from Prajit Dutta. 2001 MIT Press. Chapters 11 to 18.