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Game theory

Question $3$

This question is about extensive$-$form $($EF$)$ games. Recall that an EF game is played on a tree, where the vertices are partitioned between the players. The game proceeds as follows. We place a token on the root. In each turn, the player who controls the vertex on which the token is placed, chooses a child. The game ends once a leaf is reached. Each leaf is labeled by a utility for each player. More formally, a strategy for Player $i$ is ${}_i$:$V_{i}V,$ the outcome of a profile, denoted out $({}_1,$dots,${}_k),$ is a leaf, and payoff ${?}_i({}_1,$dots,${}_k)$ is the payoff of Player $i$ in the leaf out $({}_1,$dots,${}_k).$

For an EF game $G,$ define the social optimum as the leaf OPT $(G)=ma{x}_{\text{l}\text{e}\text{a}\text{f}l}{\displaystyle \underset{1ik}{\overset{?}{}}}payof{f}_i(l).$ Recall that we saw that every EF game has at least one NE $($in fact, an SPE$).$ Define payoff $(P)={\displaystyle \underset{1ik}{\overset{?}{}}}$ payoff ${?}_i(P).$ Show that the PoA in EF games is unbounded: for every minN, there is a game $G$ such that

$\frac{\text{O}\text{P}\text{T}(G)}{mi{n}_{\text{N}\text{E}\text{P}}\text{p}\text{a}\text{y}\text{o}\text{f}\text{f}(P)}>m$

Recall the BRD algorithm that we saw in class: consider a profile $P.$ For each Player i check whether there is a strategy ${}_i^{\text{'}}$ such that payoff $P$:$=P[ilarr{}^{\text{'}}]P_i(P).Ifso,$ set $P$:$=P[ilarr{}^{\text{'}}]$ and continue. $Ifno$ such player exists, $Pis$ a Nash equilibrium.

Describe $anEF$ game $in$ which $BRD$ never terminates. $(Note$ that this $is$ somewhat surprising since $anNE$ exists $in$ every $EF$ game$).$

$1$