Decision Making using Game Theory: An introduction for managers

66Sequential decision making and cooperative games of strategywhere O c denotes the complementary event of O, i.e. the school fails itsOfsted inspection.p(C/O)p(O)p(O/C) p(C/O)p(O) p(C/O c )p(O c )0.75 0.620.75 0.62 0.17 0.38 0.878So schools in special measures have a 87.8% chance of passing theirOfsted inspections after closure and re-opening.Sequential decision making in two-player and multi-player gamesThe discussion so far has centred on decision making by single players.The pay-oVs have therefore depended only on the decisions taken byone individual (and nature, in the cases where there is uncertainty).Games which involve more than one player, but in which moves arestill made one after the other, are now considered.Multi-person sequential games can be thought of as decisionmakinggames played in stages and, as such, they form a bridge betweenthe decision-making theory discussed above and the ‘true’ gamesdiscussed in later chapters. To understand their main features, somefamiliar concepts need to be extended.A tree is a multi-player game tree for n players if each decision node(i.e. non-terminal node) belongs to one and only one player. Thereshould be a pay-oV at each terminal node for each player (see Figure4.11), although it may happen that some players do not own anydecision nodes. (Players cannot own terminal nodes.)The player who owns the root node (P 3 ) chooses a strategy (rb say)to start the game. This brings the decision-making process to anothernode, b, owned by another player (P 4 ), assuming it is not a terminalnode. This player in turn chooses a strategy (bg say) which leads tonode g and another player (P 1 ), who chooses a strategy (gk say) whichleads to the terminal node, k. The pay-oVs at k are, say:4 for player P 1 ; 2 for P 2 ; 1 for P 3 ; 7 for P 4 .Two or more nodes of a game tree are said to be equivalent if: