text:
Magna Graecia

Magna Graecia is a term that was used for the Greek-speaking areas of Southern Italy, in the present-day Italian regions of Calabria, Apulia, Basilicata, Campania and Sicily; these regions were extensively populated by Greek settlers starting from the 8th century BC.

The settlements in this region, founded initially by their metropoleis (mother cities), eventually evolved into strong Greek city-states (poleis), functioning independently. The settlers brought with them their Hellenic civilization, and developed their own civilisation of the highest level, due to the distance from the motherland and the influence of the indigenous peoples of southern Italy which left a lasting imprint on Italy (such as in the culture of ancient Rome). They also influenced the native peoples, such as the Sicels and the Oenotrians, who became hellenised after they adopted the Greek culture as their own. In some fields such as architecture and urban planning, they sometimes surpassed the mother country. The ancient inhabitants of Magna Graecia are called Italiotes and Siceliotes.

Remains of some of these Greek cities can be seen today, such as Neapolis ("New City", now Naples), Syrakousai (Syracuse), Akragas (Agrigento), Taras (Taranto), Rhegion (Reggio Calabria), and Kroton (Crotone). The most populous city of Magna Graecia was Sybaris (now Sibari) with an estimated population, from 600 BC to 510 BC, between 300,000 and 500,000.

The government of city-states was usually an aristocracy and the cities were often at war with each other.

The Second Punic War put an end to the independence of the cities of Magna Graecia, which were annexed to the Roman Republic in 205BC.

From the motherland Greece, art, literature and philosophy decisively influenced the life of the colonies. In Magna Graecia much impetus was given to culture, especially in some cities such as Taras (now Taranto). Noteworthy was the South Italian ancient Greek pottery, fabricated in Magna Graecia largely during the 4th century BC. The settlers of Magna Graecia had great successes in the Ancient Olympic Games in their homeland. Crotone's athletes won 18 titles in 25 Olympics. Although many of the Greek inhabitants of Magna Graecia were entirely Latinized during the Middle Ages, pockets of Greek culture and language remained and have survived to the present day. One example is the Griko people in Calabria (Bovesia) and Salento (Grecìa Salentina), some of whom still maintain their Greek language (Griko language) and customs. The Griko language is the last living trace of the Greek elements that once formed Magna Graecia.

Terminology

The original Greek expression Megálē Hellás (lit. 'Great[er] Greece'), later translated into Latin as Magna Graecia, is attested for the first time in a passage from the 2nd century BC by the Greek historian Polybius (written around 150 BC), where he ascribed the term to Pythagoras and his philosophical school.

Ancient authors use "Magna Graecia" to mean different parts of southern Italy,  including or excluding Sicily, Strabo and Livy being the most prominent advocates of the wider definitions. Strabo used the term to refer to the territory that had been conquered by the Greeks.

There are various hypotheses on the origin of the name Megálē Hellás. The term could be explained by the prosperity and cultural and economic splendour of the region (6th–5th century BC); notably by the Achaeans of the city of Kroton, to refer to the network of colonies they founded or controlled between the end of the 6th and mid-5th centuries at the time of the Pythagoreans.

Context
There were several reasons for the Greeks to establish overseas colonies; demographic crises (famine, overcrowding, etc.), stasis, a developing need for new commercial outlets and ports, and expulsion from their homeland after wars.

During the Archaic period, the Greek population grew beyond the capacity of the limited arable land of Greece proper, resulting in the large-scale establishment of colonies elsewhere: according to one estimate, the population of the widening area of Greek settlement increased roughly tenfold from 800 BC to 400 BC, from 800,000 to as many as $7 1/2$-10 million. This was not simply for trade, but also to found settlements. These Greek colonies were not, as Roman colonies were, dependent on their mother-city, but were independent city-states in their own right.

Another reason was the strong economic growth with the consequent overpopulation of the motherland. The terrain that some of these Greek city-states were in could not support a large city. Politics was also the reason as refugees from Greek city-states tended to settle away from these cities in the colonies.

Greeks settled outside of Greece in two distinct ways. The first was in permanent settlements founded by the Greeks, which formed as independent poleis. The second form was in what historians refer to as emporia; trading posts which were occupied by both Greeks and non-Greeks and which were primarily concerned with the manufacture and sale of goods. Examples of this latter type of settlement are found at Al Mina in the east and Pithekoussai in the west.

From about 750 BC the Greeks began 250 years of expansion, settling colonies in all directions.

Greek colonisation
According to Strabo's Geographica, the colonisation of Magna Graecia had already begun by the time of the Trojan War and lasted for several centuries.

Greeks began to settle in southern Italy in the 8th century BC. Their first great migratory wave was by the Euboeans aimed at the Gulf of Naples (Pithecusae, Cumae) and the Strait of Messina (Zancle, Rhegium). Pithecusae on the island of Ischia is considered the oldest Greek settlement in Italy, and Cumae their first colony on the mainland of Italy.

The second wave was of the Achaeans who concentrated initially on the Ionian coast (Metapontion, Poseidonia, Sybaris, Kroton), shortly before 720BC. At an unknown date between the 8th and 6th centuries BC the Athenians, of Ionian lineage, founded Scylletium (near today's Catanzaro).

With colonisation, Greek culture was exported to Italy with its dialects of the Ancient Greek language, its religious rites, and its traditions of the independent polis. An original Hellenic civilization soon developed, and later interacted with the native Italic civilisations. The most important cultural transplant was the Chalcidean/Cumaean variety of the Greek alphabet, which was adopted by the Etruscans; the Old Italic alphabet subsequently evolved into the Latin alphabet, which became the most widely used alphabet in the world.

Secondary colonisation
Over time, due to overpopulation and other political and commercial reasons, the new cities expanded their presence in Italy by founding other Greek cities; effectively expanding the Greek civilisation to the whole territory known today as Magna Graecia.

Remains of some of these Greek colonies can be seen today such as those of Neapolis ('new city', now Naples), Syracusae (Syracuse), Akragas (Agrigento), Taras (Taranto) and Rhegion (Reggio Calabria).

An intense colonisation program was undertaken by Syracuse, at the time of the tyranny of Dionysius I of Syracuse, around 387–385 BC. This phenomenon affected the entire Adriatic coast, and in particular led to the foundation in Italy of Ancon (now Ancona) and Adria; in the Dalmatian coast he saw the foundation of Issa (current Vis), Pharus (Stari Grad), Dimus (Hvar); Lissus (now Lezhë) was founded on the Albanian coast. Issa in turn then founded Tragurium (now Trogir), Corcyra Melaina (now Korčula) and Epetium (now Stobreč, a suburb of Split).

Rhegium (now Reggio Calabria) founded Pyxus (Policastro Bussentino) in Lucania; Locri founded Medma (Rosarno), Polyxena and Hipponium (Vibo Valentia) in present-day Calabria; Sybaris (now Sibari) revitalised the indigenous centres of Laüs and Scydrus in Calabria and founded Poseidonia (Paestum), in Campania; Kroton (now Crotone) founded Terina and participated in the foundation of Caulonia (near Monasterace marina) in Calabria; Messana (now Messina), in collaboration with Rhegium, founded Metaurus (Gioia Tauro); Taras together with Thurii founded Heracleia (Policoro) in Lucania in 434 BC, and also Callipolis ('beautiful city').

Expansion and conflict
At the beginning of the 6th century, all the main cities of Magna Graecia on the Ionian Sea had achieved a high economic and cultural development, which shifted their interests towards expansion of their territory by waging war on neighbouring cities. The 6th century was therefore characterised by great clashes between the colonies. Some of the clashes that established the new balance and the new relationships of force were the Battle of the Sagra river (the clash between Locri Epizefiri and Kroton), the destruction of Siris (by Sybaris and Metapontum), and the clash between Kroton and Sybaris (which ended with the destruction of the latter).

As with all the events of this period precise dates are unknown, but the destruction of Sybaris may have occurred around 510 BC, while the two other clashes are placed around 580-560 BC, with the destruction of Siris before the Battle of the Sagra.

Roman Era

The first Greek city to be absorbed into the Roman Republic was Neàpolis in 327BC.

At the beginning of the 3rd century, Rome was a great power but had not yet entered into conflict with most of Magna Graecia, which had been allies of the Samnites. However, the needs of the Roman populace determined their need for territorial expansion towards the south. As the Greek cities of southern Italy came under threat from the Bruttii and Lucanians from the end of the 4th century BC, they asked for help from Rome, which exploited this opportunity by sending military garrisons in the 280s BC.

Following Rome's victory over Taras after the Pyrrhic War in 272BC, most of the cities of southern Italy were linked to Rome with pacts and treaties (foedera) which sanctioned a sort of indirect control.

Sicily was conquered by Rome during the First Punic War. Only Syracuse remained independent until 212 because its king Hiero II was a devoted ally of the Romans. His grandson Hieronymus however allied with Hannibal, which prompted the Romans to besiege the city, which fell in 212BC.

After the second Punic War, Rome pursued an unprecedented program of reorganisation in the rest of Magna Graecia, where many of the cities were annexed to the Roman Republic in 205BC, as a consequence of their defection to Hannibal. Roman colonies (civium romanorum) were the main element of the new territorial control plan starting from the lex Atinia of 197BC. In 194BC, garrisons of 300 Roman veterans were implanted in Volturnum, Liternum, Puteoli, Salernum and Buxentum, and to Sipontum on the Adriatic. This model was replicated in the territory of the Brettii; 194BC saw the foundation of the Roman colonies of Kroton and Tempsa, followed by the Latin colonies of Copia (193BC) and Valentia (192BC).

The social, linguistic and administrative changes arising from the Roman conquest only took root in this region by the 1st century AD, while Greek culture remained strong and was actively cultivated as shown by epigraphic evidence.

Middle Ages

During the Early Middle Ages, following the disastrous Gothic War, new waves of Byzantine Christian Greeks fleeing the Slavic invasion of Peloponnese settled in Calabria, further strengthened the Hellenic element in the region. The iconoclast emperor Leo III appropriated lands that had been granted to the Papacy in southern Italy and the Eastern Roman (Byzantine) Empire continued to govern the area in the form of the Catapanate of Italy (965 -1071) through the Middle Ages, well after northern Italy fell to the Lombards.

At the time of the Normans' late medieval conquest of southern Italy and Sicily (in the late 12th century), the Salento peninsula (the "heel" of Italy), up to one-third of Sicily (concentrated in the Val Demone), and much of Calabria and Lucania were still largely Greek-speaking. Some regions of southern Italy experienced demographic shifts as Greeks began to migrate northwards in significant numbers from regions further south; one such region was Cilento, which came to have a Greek-speaking majority. At this time the language had evolved into medieval Greek, also known as Byzantine Greek, and its speakers were known as Byzantine Greeks. The resultant fusion of local Byzantine Greek culture with Norman and Arab culture (from the Arab occupation of Sicily) gave rise to Norman-Arab-Byzantine culture in Sicily.

Mainland Italy
This is a list of the 22 poleis ("city-states") in Italy, according to Mogens Herman Hansen. It does not list all the Hellenic settlements, only those organised around a polis structure.

Sicily
This is a list of the 46 poleis ("city-states") in Sicily, according to Mogens Herman Hansen. It does not list all the Hellenic settlements, only those organised around a polis structure.

Administration

The administrative organisation of Magna Graecia was inherited from the Hellenic poleis, taking up the concept of "city-states" administered by the aristocracy. The cities of Magna Graecia were independent like the Greek poleis of the motherland, and had an army and a military fleet. There were also cases of tyranny as in Syracuse, governed by the tyrant Dionysius, who fought the Carthaginians until his death.

Economy
In the cities of Magna Graecia, trade, agriculture and crafts developed. Initially oriented to the indigenous Italic populations, the trade was immediately an excellent channel of exchange with the Greeks of the motherland, even if today it is difficult to establish precisely the type of goods traded and the volume of these exchanges.

Coinage

Greek coinage of Italy and Sicily originated from local Italiotes and Siceliotes who formed numerous city-states. These Hellenistic communities descended from Greek migrants. Southern Italy was so thoroughly Hellenized that it was known as the Magna Graecia. Each of the polities struck their own coinage.

Taras (or Tarentum) was among the most prominent city-states.

By the second century BC, some of these Greek coinages evolved under Roman rule, and can be classified as the first Roman provincial currencies.

Culture
The Greek colonists of Magna Graecia elaborated a civilization of the highest level, which had peculiar characteristics, due to the distance from the motherland and the influence of the indigenous peoples of southern Italy. From the motherland Greece, art, literature and philosophy decisively influenced the life of the colonies. In Magna Graecia much impetus was given to culture, especially in some cities, such as Taras (now Taranto). Pythagoras moved to Crotone where he founded his school in 530 BC. Among others, Aeschylus, Herodotus, Xenophanes and Plato visited Magna Graecia.

Among the illustrious characters born in Magna Graecia are the philosophers Parmenides of Elea, Zeno of Elea, Gorgias of Lentini and Empedocles of Agrigento; the Pythagoreans Philolaus of Crotone, Archytas of Taranto, Lysis of Taranto, Echecrates and Timaeus of Locri; the mathematician Archimedes of Syracuse; the poets Theocritus of Syracuse, Stesichorus, Ibycus of Reggio Calabria, Nossis of Locri, Alexis of Thuri and Leonidas of Taranto; the doctors Alcmeon of Crotone and Democedes of Crotone; the sculptor from Reggio Clearchus; the painter Zeuxis, the musicologist Aristoxenus of Taranto and the legislator Zaleucus of Locri.

Language
A remnant of Greek influence can be found in the survival of the Greek language in some villages of the above-mentioned Salento peninsula (the "heel" of Italy). This living dialect of Greek, known locally as Griko, is found in the Italian regions of Calabria and Apulia. Griko is considered by linguists to be a descendant of Byzantine Greek, which had been the majority language of Salento through the Middle Ages, combining also some ancient Doric and local romance elements. There is a rich oral tradition and Griko folklore, limited now but once numerous, to around 30,000 people, most of them having abandoned their language in favour of Italian. Some scholars, such as Gerhard Rohlfs, argue that the origins of Griko may ultimately be traced to the colonies of Magna Graecia.

Art and architecture

Magna Graecia, in some fields such as architecture and urban planning, sometimes surpassed the mother country and the other Greek colonies. In Magna Graecia, as well as in the other Greek colonies, the Doric style enriched with showy decorations was adopted as the dominant architectural style. In Magna Graecia, in particular, a Doric style influenced by the Ionic one was also used, especially in Sicily in the Achaean colonies. In Magna Graecia, limestone was used as a building material due to the difficulty in finding other materials. The Doric style in Magna Graecia reached its apogee, surpassing that of the motherland and the other Greek colonies.

Regarding urban planning, the cities of Magna Graecia, as well as many cities of Greek colonies in other regions, were more orderly and rational in the distribution of spaces than those of the mother country, making the urban fabric more practical. The first examples of urbanistically more rational Greek cities belonged to Magna Graecia, in this case Taranto, Metapontum and Megara Hyblaea. Characteristic of this new urban concept, which later spread also in the motherland to Rhodes and Miletus, was a checkerboard road network.

In Magna Graecia painting and sculpture also reached a notable level of quality. In Magna Graecia there were examples of excellence in sculpture, coroplastics and bronzes. As for vase painting, many famous Athenian potters moved to Magna Graecia creating works influenced by the culture of the place, making their paintings peculiar and different from those of the motherland, giving rise to the South Italian ancient Greek pottery. Also noteworthy are the mosaics, the goldsmith's art and wall painting.

Noteworthy sculptures from Magna Graecia are the Apollo of Gaza, the Apollo of Piombino, the Dancing Satyr of Mazara del Vallo, the Head of a Philosopher and the Riace bronzes, while notable vases from Magna Graecia are the Darius Vase and the Nestor's Cup. Noteworthy temples of Magna Graecia are the Temple of Concordia, Agrigento, the Temple of Hera Lacinia, the Temple of Heracles, Agrigento, The Temple of Juno in Agrigento, the Temple of Olympian Zeus, Agrigento, the Temple of Apollo (Syracuse), the Temple of Athena (Syracuse), the Temple of Athena (Paestum), the Temple C (Selinus), the Temple E (Selinus), the Temple F (Selinus), the Temple of Juno Lacinia (Crotone), the Second Temple of Hera (Paestum), the Heraion at Foce del Sele, the Temple of Poseidon (Taranto), the Tavole Palatine and the Temple of Victory (Himera).

Theatre

The Sicilian Greek colonists in Magna Graecia, but also from Campania and Apulia, also brought theatrical art from their motherland. The Greek Theatre of Syracuse, the Greek Theatre of Segesta, the Greek Theatre of Tindari, the Greek Theatre of Hippana, the Greek Theatre of Akrai, the Greek Theatre of Monte Jato, the Greek Theatre of Morgantina and the most famous Greek Theater of Taormina, amply demonstrate this.

Only fragments of original dramaturgical works are left, but the tragedies of the three great giants Aeschylus, Sophocles and Euripides and the comedies of Aristophanes are known.

Some famous playwrights in the Greek language came directly from Magna Graecia. Others, such as Aeschylus and Epicharmus, worked for a long time in Sicily. Epicharmus can be considered Syracusan in all respects, having worked all his life with the tyrants of Syracuse. His comedy preceded that of the more famous Aristophanes by staging the gods for the first time in comedy. While Aeschylus, after a long stay in the Sicilian colonies, died in Sicily in the colony of Gela in 456 BC. Epicarmus and Phormis, both of 6th century BC, are the basis, for Aristotle, of the invention of the Greek comedy, as he says in his book on Poetics:

"As for the composition of the stories (Epicharmus and Phormis) it came in the beginning from Sicily"

Other native dramatic authors of Magna Graecia, in addition to the Syracusan Phormis mentioned, are Achaeus of Syracuse, Apollodorus of Gela, Philemon of Syracuse and his son Philemon the younger. From Calabria, precisely from the colony of Thurii, came the playwright Alexis. While Rhinthon, although Sicilian from Syracuse, worked almost exclusively for the colony of Taranto.

Sport

The colonies sent athletes of all disciplines to the Ancient Olympic Games which were periodically held at Olympia and Delphi in Greece.

The colonists of Magna Graecia were very fond of the Hellenic games where they could prove to the Greeks that they belonged to the same place of origin, their physical strength and skills in the games were also played by their ancestors dozens of generations earlier. And for this reason the greatest sovereigns demanded that teams be trained to be sent to Greece.

Sport was therefore a channel of communication with the Hellenic peninsula, a means by which the colonies of Magna Graecia showed themselves to the rest of the Hellenic world. The settlers of Magna Graecia had great success in sporting competitions in their homeland. Crotone's athletes won 18 titles in 25 Olympics.

Essential timeline

 * 8th century BC: the first historical colony of Magna Graecia was that of Pithekoussai (current island of Ischia) founded in the 8th century BC by settlers from Chalcis and Eretria in Euboea. Probably, the island settlement of Pithekoussai was only a commercial establishment where the Greeks dealt with other peoples, especially with the Phoenician merchants, even if the issue is controversial.
 * 720 BC: the first Greek colony in mainland Italy, Kyme, is founded.
 * 7th–6th century BC: maximum splendor of Sibari.
 * 6th century BC: maximum splendor of Crotone.
 * 6th–3rd century BC: minting of coins by the cities of Magna Graecia.
 * 6th–5th century BC: maximum splendour of Magna Graecia due to the Pythagorean reforms and institutions.
 * 510 BC: Sibari was defeated by Crotone whose troops were commanded by the famous athlete Milo of Croton. The city of Sibari was destroyed and its population was condemned to exile.
 * 5th century BC: maximum splendor of Syracuse.
 * 480 BC: Gelon, tyrant of Syracuse, defeated the troops of Carthage at Himera, in the north of Sicily.
 * 474 BC: The fleet led by Hiero I, tyrant of Syracuse, assisted Kyme threatened by the Etruscans. This victory marked the end of the Etruscan extension in Campania.
 * 459–454 BC: after an internal civil war in Crotone, the cities of Magna Graecia once linked to it, dissolve the bond of subjection.
 * 444–443 BC: foundation of Thourioi. An Athenian expedition, officially Panhellenic because it was made up of Greeks from the islands of the Aegean Sea, founded the city of Thourioi. In reality, the cities of the Aegean Sea were part of the Delian League, a military league under the rule of Athens. The city of Thourioi hosted important people such as Herodotus, Protagoras, Hippodamus of Miletus and Lysias.
 * 415–413 BC: The Sicilian Expedition occurred. It was an Athenian military expedition to Sicily, which took place from 415 to 413 BC, during the Peloponnesian War, between Athens on one side and Sparta, Syracuse and Corinth on the other. The expedition ended in a devastating defeat for the Athenian forces, severely impacting Athens. After the first Athenian victories, which put the Syracusan army in serious difficulty, the tide of the war was turned upside-down due to the Spartan reinforcements under the command of Gylippus. The defeat of the Athenian army led to the imprisonment of its soldiers in the Syracusan latomies, where they were forced to live in hardship and suffering until their death; few were the survivors who managed to return to their homeland. The failure of the expedition marked the beginning of the military and political decline of Athens, followed by the aristocratic coup d'état of 411 BC; it also marked Athens' definitive defeat in the Peloponnesian War (404 BC). Thucydides, an Athenian historian, dedicates two books of his work History of the Peloponnesian War to the Athenian expedition, to underline the magnitude and exceptionality of the event. Thus he began "a new work, a work on Sicily" which became the background of the Peloponnesian War (431–404 BC). The Parallel Lives of Plutarch (in particular the Life of Nicias) and the Bibliotheca historica of Diodorus Siculus are other important sources on the expedition to Sicily.
 * 400 BC: the cities of Magna Graecia overlooking the Tyrrhenian Sea begin at the hands of the Italic peoples.
 * 4th century BC: the cultural decline of the cities of Magna Graecia begins.
 * 387 BC: Reggio is destroyed by Syracuse.
 * 303 BC: peace between Taranto and Lucanians, who had attempted to conquer the city.
 * 285 BC: Roman garrisons settle in Thourioi.
 * 282–272 BC: Taranto was conquered by the Romans despite the intervention of Pyrrhus (Pyrrhic War in Italy).
 * 264–241 BC: First Punic War, Rome takes control of Sicily, with the exception of Syracuse, which becomes Rome's ally.
 * 215–205 BC: during the Second Punic War Syracuse and then Taranto sided with Carthage. The two cities were conquered by the Romans in 211 after a three-year siege. These events put an end to the independence of all the cities of Magna Graecia, which were annexed to the Roman Republic in 205 BC.

Modern and contemporary Italy

Greek nobles started taking refuge in Italy following the Fall of Constantinople in 1453. Greeks immigrated once again to the region in the 16th and 17th centuries in reaction to the conquest of the Peloponnese by the Ottoman Empire. Especially after the end of the Siege of Coron (1534), large numbers of Greeks took refuge in the areas of Calabria, Salento and Sicily. Greeks from Coroni, the so-called Coronians, were nobles, who brought with them substantial movable property.

Other Greeks who moved to Italy came from the Mani Peninsula of the Peloponnese. The Maniots (their name originating from the Greek word mania) were known for their proud military traditions and for their bloody vendettas, many of which continue today. Another group of Maniot Greeks moved to Corsica in the 17th century under the protection of the Republic of Genoa.

Although many of the Greek inhabitants of Magna Graecia were entirely Latinized during the Middle Ages, pockets of Greek culture and language remained and have survived to the present day. One example is the Griko people in Calabria (Bovesia) and Salento (Grecìa Salentina), some of whom still maintain their Greek language (Griko language) and customs. The Griko language is the last living trace of the Greek elements that once formed Magna Graecia. Their working practices have been passed down through generations through storytelling and allowing the observation of work. The Italian parliament recognizes the Griko people as an ethnolinguistic minority under the official name of Minoranze linguistiche Grike dell'Etnia Griko-Calabrese e Salentina.

Messina in Sicily is home to a small Greek-speaking minority, which arrived from the Peloponnese between 1533 and 1534 when fleeing the expansion of the Ottoman Empire. They were officially recognised in 2012.

Valle dei Templi

The Valle dei Templi, or Valley of the Temples, is an archaeological site in Agrigento (ancient Greek Akragas), Sicily. It is one of the most outstanding examples of ancient Greek art and architecture of Magna Graecia. The term "valley" is a misnomer, the site is located on a ridge outside the town of Agrigento.

Since 1997, the entire area has been included in the UNESCO World Heritage List. The archaeological and landscape park of the Valle dei Templi, with its 1,300 hectares, is the largest archaeological park in Europe and the Mediterranean basin.

The Valley includes remains of seven temples, all in Doric style. The ascription of the names, apart from that of the Olympeion, is a mere tradition established in Renaissance times. The temples are:

 * Temple of Concordia, whose name comes from a Latin inscription found nearby, and which was built in the 5th century BC. Turned into a church in the 6th century AD, it is now one of the best preserved in the Valley.
 * Temple of Juno, also built in the 5th century BC. It was burnt in 406 BC by the Carthaginians.
 * Temple of Heracles, who was one of the most venerated deities in the ancient Akragas. It is the most ancient in the Valley: destroyed by an earthquake, it consists today of only eight columns.
 * Temple of Olympian Zeus, built in 480 BC to celebrate the city-state's victory over Carthage. It is characterized by the use of large-scale atlases.
 * Temple of Castor and Pollux. Despite its remains including only four columns, it is now the symbol of modern Agrigento.
 * Temple of Hephaestus (Vulcan), also dating from the 5th century BC. It is thought to have been one of the most imposing constructions in the valley; it is now however one of the most eroded.
 * Temple of Asclepius, located far from the ancient town's walls; it was the goal of pilgrims seeking cures for illness.

The Valley is also home to the so-called Tomb of Theron, a large tuff monument of pyramidal shape; scholars suppose it was built to commemorate the Romans killed in the Second Punic War.

Poseidonia and Elea

Cilento, Vallo di Diano and Alburni National Park (Italian Parco Nazionale del Cilento, Vallo di Diano e Alburni) is an Italian national park in the Province of Salerno, in Campania in southern Italy. It includes much of the Cilento, the Vallo di Diano and the Monti Alburni. It was founded in 1991, and was formerly known as the Parco Nazionale del Cilento e Vallo di Diano. In 1998 it became a World Heritage Site of UNESCO, along with the ancient Greek towns of Poseidonia, Elea and the Padula Charterhouse.

Much of the most celebrated features of the Poseidonia site today are the three large temples in the Archaic version of the Greek Doric order, dating from about 550 to 450 BC. All are typical of the period, with massive colonnades having a very pronounced entasis (widening as they go down), and very wide capitals resembling upturned mushrooms. Above the columns, only the second Temple of Hera retains most of its entablature, the other two having only the architrave in place. These were dedicated to Hera and Athena (Juno and Minerva to the Romans), although previously they often have been identified otherwise, following eighteenth-century arguments. The two temples of Hera are right next to each other, while the Temple of Athena is on the other side of the town centre. There were other temples, both Greek and Roman, which are far less well preserved.

Remains of Elea walls, with traces of one gate and several towers, of a total length of over three miles, still exist, and belong to three different periods, in all of which the crystalline limestone of the locality is used. Bricks were also employed in later times; their form is peculiar to this place, each having two rectangular channels on one side, and being about 1.5 inches square, with a thickness of nearly 4 inches They all bear Greek brick-stamps. There are some remains of cisterns on the site, and, various other traces of buildings.

Syracuse

Syracuse was founded in 733 BC by Greek settlers from Corinth and Tenea, led by the oecist (colonizer) Archias. There are many attested variants of the name of the city including Συράκουσαι Syrakousai, Συράκοσαι Syrakosai and Συρακώ Syrakō. In the modern day, the city is listed by UNESCO as a World Heritage Site along with the Necropolis of Pantalica.

The buildings of Syracuse from the Greek period are:
 * The city walls
 * The Temple of Apollo, at Piazza Emanuele Pancali, adapted to a church in Byzantine times and to a mosque under Arab rule.
 * The Fountain of Arethusa, on Ortygia island. According to a legend, the nymph Arethusa, hunted by Alpheus, was turned into a spring by Artemis and appeared here.
 * The Greek Theatre of Syracuse, whose cavea is one of the largest ever built by the ancient Greeks: it has 67 rows, divided into nine sections with eight aisles. Only traces of the scene and the orchestra remain. The edifice (still used today) was modified by the Romans, who adapted it to their different style of spectacles, including circus games. Near the theatre are the latomìe, stone quarries, also used as prisons in ancient times. The most famous latomìa is the Orecchio di Dionisio ("Ear of Dionysius").
 * The Tomb of Archimede, in the Grotticelli Necropolis. Decorated with two Doric columns.
 * The Temple of Olympian Zeus, about 3 km outside the city, built around the 6th century BC.

Apulia

 * Rudiae
 * Siponto

Basilicata

 * Heraclea Lucania
 * Metapontum
 * Tavole Palatine

Calabria

 * Caulonia
 * Krimisa
 * Laüs
 * Skylletion
 * Sybaris
 * Terina
 * Thurii
 * Timpone della Motta

Campania

 * Cumae
 * Elea
 * Heraion at Foce del Sele
 * Poseidonia

Sicily

 * Abacaenum
 * Adranon
 * Aetna
 * Akrai
 * Akrillai
 * Apollonia
 * Casmenae
 * Eryx
 * Halaesa
 * Helorus
 * Heraclea Minoa
 * Herbessos
 * Himera
 * Hybla Gereatis
 * Maktorion
 * Megara Hyblaea
 * Monte Adranone
 * Morgantina
 * Motya
 * Naxos
 * Segesta
 * Selinunte
 * Tindari
 * Valle dei Templi

title: Magna Graecia


text:
European Language Council

The European Language Council is a permanent and independent association whose main aim is the quantitative and qualitative improvement of knowledge of the languages and cultures of the European Union and beyond. Membership is open to all institutions of higher education and all national and international associations with a special interest in languages.

The association's permanent secretariat is at Freie Universität in Berlin, Germany.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: European Language Council


text:
Pope Paschal I

Pope Paschal I (Paschalis I; died 824) was the bishop of Rome and ruler of the Papal States from 25 January 817 to his death in 824.

Paschal was a member of an aristocratic Roman family. Before his election to the papacy, he was abbot of St. Stephen's monastery, which served pilgrims. In Rome in 823 he crowned Lothair I as Holy Roman Emperor. He rebuilt a number of churches in Rome, including three basilicas.

Early life
According to the Liber Pontificalis, Paschal was a native of Lazio Rome born Pascale Massimi and son of Bonosus and Episcopa Theodora. The Liber Censuum says that Paschal was from the Massimi branch of the Massimo family, as was his predecessor, Stephen IV.

Pope Leo III placed Paschal in charge of the monastery of St Stephen of the Abyssinians, where his responsibilities included the care of pilgrims visiting Rome. According to early modern accounts, Leo III may have elevated Paschal as the cardinal priest of Santa Prassede. Goodson attributes this account to a "desire to explain the attention that the pope so lavishly and prominently paid to that church later in his career."

Accession
Paschal became pope on 25 January 817, just one day after the sudden death of Stephen IV. This decision was made without the sanction of Emperor Louis the Pious. Paschal began his pontificate apologizing for this slight, stressing that the office had been thrust upon him. He claimed that the decision had been made to avoid factional strife in Rome. According to the Liber Pontificalis, Pope Paschal's legate Theodore returned with a document titled Pactum cum Paschali pontifice, in which the emperor congratulated Paschal, recognized his sovereignty over the Papal States, and guaranteed the free election of future pontiffs. This document has since been challenged by historians as a forgery.

Papacy
At first, Emperor Louis confirmed the agreement reached in Rheims with Paschal's predecessor, Stephen IV, and detailed in the document Pactum Ludovicanum about free papal elections and noninterference in Church affairs unless officially asked for help. The two worked together to send Archbishop Ebbo of Rheims to evangelize the Danes in 822.

On Easter Sunday of 823, Paschal crowned and anointed Louis's son Lothair I. Lothair was less amenable to cooperating with the Papal Curia than his father. He held a court and declared Farfa Abbey, just north of Rome, exempt from papal taxation. Paschal's aristocratic opponents in the papal palace, especially his former legate, Theodore, and his son-in-law, Leo, who turned to the young leader of the Franks for support in their opposition to Paschal. The decision outraged the Roman nobility and led to an uprising against the authority of the Roman Curia in northern Italy led by Theodore and Leo. The revolt was quickly suppressed, and two of its leaders were seized, blinded, and afterwards beheaded by members of the papal household. Paschal denied any involvement, but the Emperor remained suspicious and sent two commissioners to investigate. Paschal refused to submit to the authority of the imperial court, but he did take an oath of purgation before a synod of thirty-four bishops. The commissioners returned to Aachen, and Emperor Louis let the matter drop.

Construction projects
Paschal gave shelter to exiled monks from the Byzantine Empire who had fled persecution for their opposition to iconoclasm. He both offered the exiled Byzantine mosaic artists work decorating churches in Rome and wrote to Louis the Pious and the Byzantine emperor Leo the Armenian in support of those who opposed iconoclasm.

Paschal rebuilt three basilicas of Rome: Santa Prassede, Santa Maria in Domnica, and Santa Cecilia in Trastevere. These churches contain mosaics with lifelike portraits of Paschal. Paschal is credited with finding the body of Saint Cecilia in the Catacomb of Callixtus and translating it to the rebuild the basilica of Santa Cecilia in Trastevere. Paschal also undertook significant renovations on Basilica di Santa Maria Maggiore. In addition, Paschal added two oratories to Old St. Peter's Basilica, SS. Processus et Martinianus and SS. Xistus et Fabianus, which did not survive the 16th century renovation of St. Peter's.

Paschal is also sometimes credited with the renovation of Santo Stefano del Cacco in early modern sources, but this renovation was actually undertaken by Pope Paschal II.

According to Goodson, Paschal "used church-building to express the authority of the papacy as an independent state."

Writings

Only six known letters written by Paschal remain. The first (Jaffé 2546) confirms the possessions of the Territorial Abbey of Farfa. The second and third (Jaffé 2547 and Jaffé 2548) were written to a Frankish abbot prior to and after his elevation as archbishop of Vienne. The fourth (Jaffé 2550) was written to Louis the Pious. The fifth (Jaffé 2551, preserved in the Biblioteca Ambrosiana) confirms the privileges of the church of Ravenna. The last (Jaffé 2553) was written to Ebbo, the archbishop of Reims.

Death
Paschal died on 11 February 824. The Roman Curia refused him the honour of burial within St. Peter's Basilica because of his harsh government of the Roman people. He was instead buried in the Basilica of Santa Prassede, which also contains the famous Episcopa Theodora mosaic of his mother.

Paschal was canonized in the late sixteenth century. His feast day in the Roman calendar prior to 1963 was 14 May. It is currently celebrated on 11 February.

Pope and Patriarch of Rome (817–824), confessor is honored in Orthodox Church on 14 May.

title: Pope Paschal I


text:
Nash equilibrium

In game theory, the Nash equilibrium is the most commonly-used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly.

If each player has chosen a strategy – an action plan based on what has happened so far in the game –  and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium.

If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A. In a game in which Carol and Dan are also players, (A, B, C, D) is a Nash equilibrium if A is Alice's best response to (B, C, D), B is Bob's best response to (A, C, D), and so forth.

Nash showed that there is a Nash equilibrium, possibly in mixed strategies, for every finite game.

Applications
Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers. In a strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as their own. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do taking into account what the player expects the others to do. Nash equilibrium requires that one's choices be consistent: no players wish to undo their decision given what the others are deciding.

The concept has been used to analyze hostile situations such as wars and arms races (see prisoner's dilemma), and also how conflict may be mitigated by repeated interaction (see tit-for-tat). It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see stag hunt). It has been used to study the adoption of technical standards, and also the occurrence of bank runs and currency crises (see coordination game). Other applications include traffic flow (see Wardrop's principle), how to organize auctions (see auction theory), the outcome of efforts exerted by multiple parties in the education process, regulatory legislation such as environmental regulations (see tragedy of the commons), natural resource management, analysing strategies in marketing, penalty kicks in football (see matching pennies), robot navigation in crowds, energy systems, transportation systems, evacuation problems and wireless communications.

History
Nash equilibrium is named after American mathematician John Forbes Nash Jr. The same idea was used in a particular application in 1838 by Antoine Augustin Cournot in his theory of oligopoly. In Cournot's theory, each of several firms choose how much output to produce to maximize its profit. The best output for one firm depends on the outputs of the others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure-strategy Nash equilibrium. Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally.

The modern concept of Nash equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible pure strategies (which might put 100% of the probability on one pure strategy; such pure strategies are a subset of mixed strategies). The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior, but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions. The contribution of Nash in his 1951 article "Non-Cooperative Games" was to define a mixed-strategy Nash equilibrium for any game with a finite set of actions and prove that at least one (mixed-strategy) Nash equilibrium must exist in such a game. The key to Nash's ability to prove existence far more generally than von Neumann lay in his definition of equilibrium. According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes [their] payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others." Putting the problem in this framework allowed Nash to employ the Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose.

Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make a unique prediction. They have proposed many solution concepts ('refinements' of Nash equilibria) designed to rule out implausible Nash equilibria. One particularly important issue is that some Nash equilibria may be based on threats that are not 'credible'. In 1965 Reinhard Selten proposed subgame perfect equilibrium as a refinement that eliminates equilibria which depend on non-credible threats. Other extensions of the Nash equilibrium concept have addressed what happens if a game is repeated, or what happens if a game is played in the absence of complete information. However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others.

Nash equilibrium
A strategy profile is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?"

For instance if a player prefers "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to the other players' strategies in that equilibrium.

Formally, let $$S_i$$ be the set of all possible strategies for player $$i$$, where $$i = 1, \ldots, N$$. Let $$s^* = (s_i^*, s_{-i}^*)$$ be a strategy profile, a set consisting of one strategy for each player, where $$s_{-i}^*$$ denotes the $$N-1$$ strategies of all the players except $$i$$. Let $$u_i(s_i, s_{-i}^*)$$ be player i's payoff as a function of the strategies. The strategy profile $$s^*$$ is a Nash equilibrium if
 * $$u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*) \;\;{\rm for \; all}\;\; s_i \in S_i$$

A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be weak: a player might be indifferent among several strategies given the other players' choices. It is unique and called a strict Nash equilibrium if the inequality is strict so one strategy is the unique best response:
 * $$u_i(s_i^*, s_{-i}^*)> u_i(s_i, s_{-i}^*) \;\;{\rm for \; all}\;\; s_i \in S_i, s_i \neq s_i^*$$

The strategy set $$S_i$$ can be different for different players, and its elements can be a variety of mathematical objects. Most simply, a player might choose between two strategies, e.g. $$S_i = \{\text{Yes}, \text{No}\}.$$ Or, the strategy set might be a finite set of conditional strategies responding to other players, e.g. $$S_i = \{\text{Yes}|p=\text{Low}, \text{No}|p=\text{High}\}.$$ Or, it might be an infinite set, a continuum or unbounded, e.g. $$S_i = \{\text{Price}\}$$ such that $$\text{Price}$$ is a non-negative real number. Nash's existing proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it.

Pure/mixed equilibrium
A game can have a pure-strategy or a mixed-strategy Nash equilibrium. In the latter, a pure strategy is chosen stochastically with a fixed probability.

Strict/Non-strict equilibrium
Suppose that in the Nash equilibrium, each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?"

If every player's answer is "Yes", then the equilibrium is classified as a strict Nash equilibrium.

If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives exactly the same payout (i.e. the player is indifferent between switching and not), then the equilibrium is classified as a weak or non-strict Nash equilibrium.

Equilibria for coalitions
The Nash equilibrium defines stability only in terms of individual player deviations. In cooperative games such a concept is not convincing enough. Strong Nash equilibrium allows for deviations by every conceivable coalition. Formally, a strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be Pareto efficient. As a result of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium.

A refined Nash equilibrium known as coalition-proof Nash equilibrium (CPNE) occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate. Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the theory of the core.

Nash's existence theorem
Nash proved that if mixed strategies (where a player chooses probabilities of using various pure strategies) are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium, which might be a pure strategy for each player or might be a probability distribution over strategies for each player.

Nash equilibria need not exist if the set of choices is infinite and non-compact. For example:

 * A game where two players simultaneously name a number and the player naming the larger number wins does not have a NE, as the set of choices is not compact because it is unbounded.
 * Each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists (if the number could equal 5, the Nash equilibrium would have both players choosing 5 and tying the game). Here, the set of choices is not compact because it is not closed.

However, a Nash equilibrium exists if the set of choices is compact with each player's payoff continuous in the strategies of all the players.

Rosen's existence theorem
Rosen extended Nash's existence theorem in several ways. He considers an n-player game, in which the strategy of each player i is a vector si in the Euclidean space Rmi. Denote m:=m1+...+mn; so a strategy-tuple is a vector in Rm. Part of the definition of a game is a subset S of Rm such that the strategy-tuple must be in S. This means that the actions of players may potentially be constrained based on actions of other players. A common special case of the model is when S is a Cartesian product of convex sets S1,...,Sn, such that the strategy of player i must be in Si. This represents the case that the actions of each player i are constrained independently of other players' actions. If the following conditions hold:

 * T is convex, closed and bounded;
 * Each payoff function ui is continuous in the strategies of all players, and concave in si for every fixed value of s-i.

Then a Nash equilibrium exists. The proof uses the Kakutani fixed-point theorem. Rosen also proves that, under certain technical conditions which include strict concavity, the equilibrium is unique.

Nash's result refers to the special case in which each Si is a simplex (representing all possible mixtures of pure strategies), and the payoff functions of all players are bilinear functions of the strategies.

Rationality
The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal.

Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with threats they would not actually carry out. For such games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis.

Coordination game
The coordination game is a classic two-player, two-strategy game, as shown in the example payoff matrix to the right. There are two pure-strategy equilibria, (A,A) with payoff 4 for each player and (B,B) with payoff 2 for each. The combination (B,B) is a Nash equilibrium because if either player unilaterally changes their strategy from B to A, their payoff will fall from 2 to 1.

A famous example of a coordination game is the stag hunt. Two players may choose to hunt a stag or a rabbit, the stag providing more meat (4 utility units, 2 for each player) than the rabbit (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, the stag hunter will totally fail, for a payoff of 0, whereas the rabbit hunter will succeed, for a payoff of 1. The game has two equilibria, (stag, stag) and (rabbit, rabbit), because a player's optimal strategy depends on their expectation on what the other player will do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they think the other will hunt the rabbit, they too will hunt the rabbit. This game is used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation.

Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:

In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player are (50%, 50%).

Network traffic
An application of Nash equilibria is in determining the expected flow of traffic in a network. Consider the graph on the right. If we assume that there are $$x$$ "cars" traveling from $A$ to $D$, what is the expected distribution of traffic in the network?

This situation can be modeled as a "game", where every traveler has a choice of 3 strategies and where each strategy is a route from $A$ to $D$ (one of $ABD$, $ABCD$, or $ACD$). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via $ABD$ experiences travel time of $$1+\frac{x}{100}+2$$, where $$x$$ is the number of cars traveling on edge $AB$. Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to their travel time. For the graph on the right, if, for example, 100 cars are travelling from $A$ to $D$, then equilibrium will occur when 25 drivers travel via $ABD$, 50 via $ABCD$, and 25 via $ACD$. Every driver now has a total travel time of 3.75 (to see this, a total of 75 cars take the $AB$ edge, and likewise, 75 cars take the $CD$ edge).

Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via $ABD$ and the other 50 through $ABCD$, then travel time for any single car would actually be 3.5, which is less than 3.75. This is also the Nash equilibrium if the path between $AB$ and $CD$ is removed, which means that adding another possible route can decrease the efficiency of the system, a phenomenon known as Braess's paradox.

Competition game
This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then they have to give up two points to the other.

This game has a unique pure-strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if the game begins at the green square, it is in player 1's interest to move to the purple square and it is in player 2's interest to move to the blue square. Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria: (0,0), (1,1), (2,2), and (3,3).

Nash equilibria in a payoff matrix
There is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the payoff pair of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium.

We can apply this rule to a 3×3 matrix:

Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash equilibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A), 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B), 25 is the maximum of the second column and 40 is the maximum of the first row; the same applies for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns.

This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair is the maximum of the row. If these conditions are met, the cell represents a Nash equilibrium. Check all columns this way to find all NE cells. An N×N matrix may have between 0 and N×N pure-strategy Nash equilibria.

Stability
The concept of stability, useful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria.

A Nash equilibrium for a mixed-strategy game is stable if a small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold:

 * 1) the player who did not change has no better strategy in the new circumstance
 * 2) the player who did change is now playing with a strictly worse strategy.

If these cases are both met, then a player with the small change in their mixed strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed.

In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed strategies with 100% probabilities are stable. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes their probabilities (which would neither benefit or damage the expectation of the player who did the change, if the other player's mixed strategy is still (50%,50%)), then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%).

Stability is crucial in practical applications of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium.

Finally in the eighties, building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept. Mertens stable equilibria satisfy both forward induction and backward induction. In a game theory context stable equilibria now usually refer to Mertens stable equilibria.

Occurrence
If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are:
 * 1) The players all will do their utmost to maximize their expected payoff as described by the game.
 * 2) The players are flawless in execution.
 * 3) The players have sufficient intelligence to deduce the solution.
 * 4) The players know the planned equilibrium strategy of all of the other players.
 * 5) The players believe that a deviation in their own strategy will not cause deviations by any other players.
 * 6) There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on.

Where the conditions are not met
Examples of game theory problems in which these conditions are not met:
 * 1) The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. For instance, the prisoner's dilemma is not a dilemma if either player is happy to be jailed indefinitely.
 * 2) Intentional or accidental imperfection in execution. For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium. Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through negation of the common knowledge criterion leading to possible victory for the player. (An example would be a player suddenly putting the car into reverse in the game of chicken, ensuring a no-loss no-win scenario).
 * 3) In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess. Or, if known, it may not be known to all players, as when playing tic-tac-toe with a small child who desperately wants to win (meeting the other criteria).
 * 4) The criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. Players wrongly distrusting each other's rationality may adopt counter-strategies to expected irrational play on their opponents’ behalf. This is a major consideration in "chicken" or an arms race, for example.

Where the conditions are met
In his Ph.D. dissertation, John Nash proposed two interpretations of his equilibrium concept, with the objective of showing how equilibrium points can be connected with observable phenomenon.

"(...) One interpretation is rationalistic: if we assume that players are rational, know the full structure of the game, the game is played just once, and there is just one Nash equilibrium, then players will play according to that equilibrium."

This idea was formalized by R. Aumann and A. Brandenburger, 1995, Epistemic Conditions for Nash Equilibrium, Econometrica, 63, 1161-1180 who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly known, then the conjectures must be a Nash equilibrium (a common prior assumption is needed for this result in general, but not in the case of two players. In this case, the conjectures need only be mutually known).

A second interpretation, that Nash referred to by the mass action interpretation, is less demanding on players:

"[i]t is unnecessary to assume that the participants have full knowledge of the total structure of the game, or the ability and inclination to go through any complex reasoning processes. What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. If there is a stable average frequency with which each pure strategy is employed by the average member of the appropriate population, then this stable average frequency constitutes a mixed strategy Nash equilibrium."

For a formal result along these lines, see Kuhn, H. and et al., 1996, "The Work of John Nash in Game Theory", Journal of Economic Theory, 69, 153–185.

Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics and evolutionary biology, the NE has explanatory power. The payoff in economics is utility (or sometimes money), and in evolutionary biology is gene transmission; both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim that strategies failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "stability" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research.

NE and non-credible threats
The Nash equilibrium is a superset of the subgame perfect Nash equilibrium. The subgame perfect equilibrium in addition to the Nash equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game. This eliminates all non-credible threats, that is, strategies that contain non-rational moves in order to make the counter-player change their strategy.

The image to the right shows a simple sequential game that illustrates the issue with subgame imperfect Nash equilibria. In this game player one chooses left(L) or right(R), which is followed by player two being called upon to be kind (K) or unkind (U) to player one, However, player two only stands to gain from being unkind if player one goes left. If player one goes right the rational player two would de facto be kind to her/him in that subgame. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) Nash equilibrium. Therefore, if rational behavior can be expected by both parties the subgame perfect Nash equilibrium may be a more meaningful solution concept when such dynamic inconsistencies arise.

Proof using the Kakutani fixed-point theorem
Nash's original proof (in his thesis) used Brouwer's fixed-point theorem (e.g., see below for a variant). This section presents a simpler proof via the Kakutani fixed-point theorem, following Nash's 1950 paper (he credits David Gale with the observation that such a simplification is possible).

To prove the existence of a Nash equilibrium, let $$r_i(\sigma_{-i})$$ be the best response of player i to the strategies of all other players.
 * $$ r_i(\sigma_{-i}) = \mathop{\underset{\sigma_i}{\operatorname{arg\,max}}} u_i (\sigma_i,\sigma_{-i}) $$

Here, $$\sigma \in \Sigma$$, where $$\Sigma = \Sigma_i \times \Sigma_{-i}$$, is a mixed-strategy profile in the set of all mixed strategies and $$ u_i $$ is the payoff function for player i. Define a set-valued function $$r\colon \Sigma \rightarrow 2^\Sigma $$ such that $$r = r_i(\sigma_{-i})\times r_{-i}(\sigma_{i}) $$. The existence of a Nash equilibrium is equivalent to $$r$$ having a fixed point.

Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied.
 * 1) $$ \Sigma$$ is compact, convex, and nonempty.
 * 2) $$r(\sigma)$$ is nonempty.
 * 3) $$r(\sigma)$$ is upper hemicontinuous
 * 4) $$r(\sigma)$$ is convex.

Condition 1. is satisfied from the fact that $$\Sigma$$ is a simplex and thus compact. Convexity follows from players' ability to mix strategies. $$\Sigma$$ is nonempty as long as players have strategies.

Condition 2. and 3. are satisfied by way of Berge's maximum theorem. Because $$ u_i $$ is continuous and compact, $$ r(\sigma_i) $$ is non-empty and upper hemicontinuous.

Condition 4. is satisfied as a result of mixed strategies. Suppose $$ \sigma_i, \sigma'_i \in r(\sigma_{-i}) $$, then $$ \lambda \sigma_i + (1-\lambda) \sigma'_i \in r(\sigma_{-i}) $$. i.e. if two strategies maximize payoffs, then a mix between the two strategies will yield the same payoff.

Therefore, there exists a fixed point in $$ r $$ and a Nash equilibrium.

When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed-point theorem." (See Nasar, 1998, p. 94.)

Alternate proof using the Brouwer fixed-point theorem
We have a game $$G=(N,A,u)$$ where $$N$$ is the number of players and $$A = A_1 \times \cdots \times A_N$$ is the action set for the players. All of the action sets $$A_i$$ are finite. Let $$\Delta = \Delta_1 \times \cdots \times \Delta_N$$ denote the set of mixed strategies for the players. The finiteness of the $$A_i$$s ensures the compactness of $$\Delta$$.

We can now define the gain functions. For a mixed strategy $$\sigma \in \Delta$$, we let the gain for player $$i$$ on action $$a \in A_i$$ be

 * $$\text{Gain}_i(\sigma,a) = \max \{0, u_i(a, \sigma_{-i}) - u_i(\sigma_{i}, \sigma_{-i})\}.$$

The gain function represents the benefit a player gets by unilaterally changing their strategy. We now define $$g = (g_1,\dotsc,g_N)$$ where

 * $$g_i(\sigma)(a) = \sigma_i(a) + \text{Gain}_i(\sigma,a)$$

for $$\sigma \in \Delta, a \in A_i$$. We see that

 * $$\sum_{a \in A_i} g_i(\sigma)(a) = \sum_{a \in A_i} \sigma_i(a) + \text{Gain}_i(\sigma,a) = 1 + \sum_{a \in A_i} \text{Gain}_i(\sigma,a) > 0.$$

Next we define:

 * $$\begin{cases} f = (f_1, \cdots, f_N) : \Delta \to \Delta \\ f_i(\sigma)(a) = \frac{g_i(\sigma)(a)}{\sum_{b \in A_i} g_i(\sigma)(b)} & a \in A_i \end{cases}$$

It is easy to see that each $$f_i$$ is a valid mixed strategy in $$\Delta_i$$. It is also easy to check that each $$f_i$$ is a continuous function of $$\sigma$$, and hence $$f$$ is a continuous function. As the cross product of a finite number of compact convex sets, $$\Delta$$ is also compact and convex. Applying the Brouwer fixed point theorem to $$f$$ and $$\Delta$$ we conclude that $$f$$ has a fixed point in $$\Delta$$, call it $$\sigma^*$$. We claim that $$\sigma^*$$ is a Nash equilibrium in $$G$$. For this purpose, it suffices to show that

 * $$ \forall i \in \{1, \cdots, N\}, \forall a \in A_i: \quad \text{Gain}_i(\sigma^*,a) = 0.$$

This simply states that each player gains no benefit by unilaterally changing their strategy, which is exactly the necessary condition for a Nash equilibrium.

Now assume that the gains are not all zero. Therefore, $$\exists i \in \{1, \cdots, N\},$$ and $$a \in A_i$$ such that $$\text{Gain}_i(\sigma^*, a) > 0$$. Then

 * $$ \sum_{a \in A_i} g_i(\sigma^*, a) = 1 + \sum_{a \in A_i} \text{Gain}_i(\sigma^*,a) > 1.$$

So let

 * $$C = \sum_{a \in A_i} g_i(\sigma^*, a).$$

Also we shall denote $$\text{Gain}(i,\cdot)$$ as the gain vector indexed by actions in $$A_i$$. Since $$\sigma^*$$ is the fixed point we have:

 * $$\begin{align}

\sigma^* = f(\sigma^*) &\Rightarrow  \sigma^*_i =  f_i(\sigma^*) \\ &\Rightarrow \sigma^*_i = \frac{g_i(\sigma^*)}{\sum_{a \in A_i} g_i(\sigma^*)(a)} \\ [6pt] &\Rightarrow \sigma^*_i = \frac{1}{C} \left (\sigma^*_i + \text{Gain}_i(\sigma^*,\cdot) \right ) \\ [6pt] &\Rightarrow C\sigma^*_i = \sigma^*_i + \text{Gain}_i(\sigma^*,\cdot) \\ &\Rightarrow \left(C-1\right)\sigma^*_i = \text{Gain}_i(\sigma^*,\cdot) \\ &\Rightarrow \sigma^*_i = \left(\frac{1}{C-1}\right)\text{Gain}_i(\sigma^*,\cdot). \end{align}$$

Since $$C > 1$$ we have that $$\sigma^*_i$$ is some positive scaling of the vector $$\text{Gain}_i(\sigma^*,\cdot)$$. Now we claim that

 * $$\forall a \in A_i: \quad \sigma^*_i(a)(u_i(a_i, \sigma^*_{-i}) - u_i(\sigma^*_i, \sigma^*_{-i})) = \sigma^*_i(a)\text{Gain}_i(\sigma^*, a) $$

To see this, first if $$\text{Gain}_i(\sigma^*, a) > 0$$ then this is true by definition of the gain function. Now assume that $$\text{Gain}_i(\sigma^*, a) = 0$$. By our previous statements we have that

 * $$\sigma^*_i(a) = \left(\frac{1}{C-1}\right)\text{Gain}_i(\sigma^*, a) = 0 $$

and so the left term is zero, giving us that the entire expression is $$0$$ as needed.

So we finally have that

 * $$\begin{align}

0 &= u_i(\sigma^*_i, \sigma^*_{-i}) - u_i(\sigma^*_i, \sigma^*_{-i}) \\ &= \left(\sum_{a \in A_i} \sigma^*_i(a)u_i(a_i, \sigma^*_{-i})\right) - u_i(\sigma^*_i, \sigma^*_{-i}) \\ & = \sum_{a \in A_i} \sigma^*_i(a) (u_i(a_i, \sigma^*_{-i}) - u_i(\sigma^*_i, \sigma^*_{-i})) \\ & = \sum_{a \in A_i} \sigma^*_i(a) \text{Gain}_i(\sigma^*, a) && \text{ by the previous statements } \\ &= \sum_{a \in A_i} \left( C -1 \right) \sigma^*_i(a)^2 > 0 \end{align}$$

where the last inequality follows since $$\sigma^*_i$$ is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore, $$\sigma^*$$ is a Nash equilibrium for $$G$$ as needed.

Computing Nash equilibria
If a player A has a dominant strategy $$s_A$$ then there exists a Nash equilibrium in which A plays $$s_A$$. In the case of two players A and B, there exists a Nash equilibrium in which A plays $$s_A$$ and B plays a best response to $$s_A$$. If $$s_A$$ is a strictly dominant strategy, A plays $$s_A$$ in all Nash equilibria. If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays their strictly dominant strategy.

In games with mixed-strategy Nash equilibria, the probability of a player choosing any particular (so pure) strategy can be computed by assigning a variable to each strategy that represents a fixed probability for choosing that strategy. In order for a player to be willing to randomize, their expected payoff for each (pure) strategy should be the same. In addition, the sum of the probabilities for each strategy of a particular player should be 1. This creates a system of equations from which the probabilities of choosing each strategy can be derived.

Examples
In the matching pennies game, player A loses a point to B if A and B play the same strategy and wins a point from B if they play different strategies. To compute the mixed-strategy Nash equilibrium, assign A the probability $$p$$ of playing H and $$(1-p)$$ of playing T, and assign B the probability $$q$$ of playing H and $$(1-q)$$ of playing T.

 * $$\begin{align}

&\mathbb{E}[\text{payoff for A playing H}] = (-1)q + (+1)(1-q) = 1 - 2q\\ &\mathbb{E}[\text{payoff for A playing T}] = (+1)q + (-1)(1-q) = 2q-1\\ &\mathbb{E}[\text{payoff for A playing H}] = \mathbb{E}[\text{payoff for A playing T}] \implies 1-2q = 2q-1 \implies q = \frac{1}{2}\\

&\mathbb{E}[\text{payoff for B playing H}] = (+1)p + (-1)(1-p) = 2p-1\\ &\mathbb{E}[\text{payoff for B playing T}] = (-1)p + (+1)(1-p) = 1-2p\\ &\mathbb{E}[\text{payoff for B playing H}] = \mathbb{E}[\text{payoff for B playing T}] \implies 2p-1 = 1-2p \implies p = \frac{1}{2}\\ \end{align}$$

Thus, a mixed-strategy Nash equilibrium in this game is for each player to randomly choose H or T with $$p = \frac{1}{2}$$ and $$q = \frac{1}{2}$$.

Oddness of equilibrium points
In 1971, Robert Wilson came up with the Oddness Theorem, which says that "almost all" finite games have a finite and odd number of Nash equilibria. In 1993, Harsanyi published an alternative proof of the result. "Almost all" here means that any game with an infinite or even number of equilibria is very special in the sense that if its payoffs were even slightly randomly perturbed, with probability one it would have an odd number of equilibria instead.

The prisoner's dilemma, for example, has one equilibrium, while the battle of the sexes has three—two pure and one mixed, and this remains true even if the payoffs change slightly. The free money game is an example of a "special" game with an even number of equilibria. In it, two players have to both vote "yes" rather than "no" to get a reward and the votes are simultaneous. There are two pure-strategy Nash equilibria, (yes, yes) and (no, no), and no mixed strategy equilibria, because the strategy "yes" weakly dominates "no". "Yes" is as good as "no" regardless of the other player's action, but if there is any chance the other player chooses "yes" then "yes" is the best reply. Under a small random perturbation of the payoffs, however, the probability that any two payoffs would remain tied, whether at 0 or some other number, is vanishingly small, and the game would have either one or three equilibria instead.

Game theory textbooks

 * Dixit, Avinash, Susan Skeath and David Reiley. Games of Strategy. W.W. Norton & Company. (Third edition in 2009.) An undergraduate text.
 * . Suitable for undergraduate and business students.
 * Fudenberg, Drew and Jean Tirole (1991) Game Theory MIT Press.
 * . Lucid and detailed introduction to game theory in an explicitly economic context.
 * Morgenstern, Oskar and John von Neumann (1947) The Theory of Games and Economic Behavior Princeton University Press.
 * . A modern introduction at the graduate level.
 * . A comprehensive reference from a computational perspective; see Chapter 3. Downloadable free online.
 * . A modern introduction at the graduate level.
 * . A comprehensive reference from a computational perspective; see Chapter 3. Downloadable free online.
 * . A modern introduction at the graduate level.
 * . A comprehensive reference from a computational perspective; see Chapter 3. Downloadable free online.

Original Nash papers

 * Nash, John (1950) "Equilibrium points in n-person games" Proceedings of the National Academy of Sciences 36(1):48-49.
 * Nash, John (1951) "Non-Cooperative Games" The Annals of Mathematics 54(2):286-295.

Other references

 * Mehlmann, A. (2000) The Game's Afoot! Game Theory in Myth and Paradox, American Mathematical Society.
 * Nasar, Sylvia (1998), A Beautiful Mind, Simon & Schuster.
 * Aviad Rubinstein: "Hardness of Approximation Between P and NP", ACM, ISBN 978-1-947487-23-9 (May 2019), DOI: https://doi.org/10.1145/3241304 . # Explains the Nash Equilibrium is a hard problem in computation.

title: Nash equilibrium


text:
Molfetta

Molfetta (Molfettese: Melfétte) is a town located in the northern side of the Metropolitan City of Bari, Apulia, southern Italy.

It has a well restored old city, and its own dialect.

History
The earliest local signs of permanent habitation are at the Neolithic site of Pulo, one of the most important such sites in southern Italy. The origins of the city can be traced to a small fishing port; antique graves testify to a fisherman's village in the fourth century BC. The position of the future city offered a valid landing to the commerce of Roman Rubo. The first indication of a toponym on the coast between Turenum (Trani) and Natiolum (Giovinazzo) is in the Itinerarium Provinciarum Antonini Augusti, edited from a third-century core. The place denominated Respa was probably a wrong transcript of the toponym Melpha, referring to a small village of fishermen.

The first official document that mentions the city dates to November 925; it documents a civitas denominated Melfi, situated on a peninsula named Sant'Andrea. The city developed under Byzantine dominion, and was later conquered by the Lombards, who included it in the Duchy of Benevento. The city repelled repeated assaults by the Saracens. As an independent seaport, Molfetta traded with other Mediterranean markets, including Venice, Alexandria, Constantinople, Syria, Amalfi and Ragusa.

At the beginning of the 11th century the Normans arrived, and the autonomy that the city preserved helped foster its development as both a commercial port with the east, and as port of embarcation for pilgrims heading to the Holy Land. The Crusades permitted the city to assume a wider importance. Among the many pilgrims was Conrad of Bavaria, who was so enamoured of the city that he became venerated as San Corrado, the protecting saint of Molfetta. During the Angevin dominion the city succeeded in remaining autonomous. However, the arrival of the Aragonese kingdom to Southern Italy, spurred turbulent struggles between French, Spanish and Italians. These wars provoked death and destruction in the whole south of Italy: the Sack of Molfetta at the hands of the French, 18–19 July 1529, was an episode that stalled the economic rebirth of the city.

In February 2006, Molfetta hosted International Youth Parliament, an event which took place the previous year in Canterbury.

Geography
Located in the north-western corner of its province, near the borders with the one of Barletta-Andria-Trani, and by the Adriatic Coast, Molfetta borders with the municipalities of Bisceglie (BT), Giovinazzo, Terlizzi and Ruvo di Puglia. The town is 27 km from Andria, 31 from Barletta and 34 from Bari.

Main sights

 * Il Pulo is one of the most important Neolithic sites of southern Italy. It is a circular cave 23 m deep with grottoes and remains of old constructions.
 * The Old Cathedral (it:Duomo di San Corrado) was built in the twelfth–thirteenth centuries in Apulian-Romanesque style, using local stone on a basilica plan, a nave with two aisles divided by four central cross-shaped pilasters. The floor has two domes.  From the apse area rise two 20 m towers, one of which acted as watchtower, the other has the usual campanile. The interior has some notable religious furnishings from the sixteenth century.
 * Several watchtowers, such as the Torre Calderina (fifteenth century) on the seaside, and the Torrione Passari, inglobated in the town's walls.
 * The New Cathedral, or church of S. Maria Assunta in Cielo, was built by the Jesuits from 1610. It houses the remains of the city's patron, San Corrado of Bavaria, in a silver reliquary bust of the saint (seventeenth century) by G. Todaro.
 * The church of San Bernardino da Siena (1451, rebuilt in 1585) includes a triptych by Duccio d'Andrea (fifteenth century) and other later paintings. Notable is the Renaissance choir. The annexed convent is now the Palazzo Civico (town hall).
 * The church of Santo Stefano, built from 1286, but with a Renaissance stone façade added in 1586.
 * Palazzo Giovene is a 16th-century palace, now used as Town Hall. It has a notable Renaissance style portal.
 * The church of Santa Maria Consolatrice degli Afflitti, simply known as Chiesa del Purgatorio, dating from 1643 and consecrated in 1667. The façade has statues representing Sts. Stephen, Peter, Paul and Lawrence and, on the two side summits, those of St. Joaquim and St. Anne. The interior houses paintings by Bernardo Cavallino and native-son Corrado Giaquinto.
 * The church of San Pietro Apostolo, simply called "San Pietro's Church", just existing in the twelfth century, but with the actual Baroque façade and bell tower, situated in the old town, at the begin of the street of the some name, near the Municipio Square.
 * The Temple of Calvary, a small Neo-Gothic construction built in 1856 and designed by the local architect Corrado De Judicibus.
 * Two km outside the city in the direction of Bisceglie, is the basilica-sanctuary of the Madonna dei Martiri. The current nave of the church is partially built over the old eleventh-century church, of which only a dome and the underlying structure remain, in today's altar area. Annexed is the Crusaders Hospital, also from the eleventh century. The basilica conserves an image that was a votive gift of some Crusaders in 1188.
 * The Molfetta Lighthouse on the eastern pier; still active today.

People
Personalities from Molfetta include Cardinal Angelo Amato, Rossella Biscotti (artist), Luigi Capotorti (19th-century composer), Leonardo Andriani (immigrant), Corrado Giaquinto (Rococo painter), Domenico Leccisi (notorious for stealing Mussolini's corpse), Riccardo Muti (conductor), Girolamo Minervini (assassinated magistrate), Caparezza (rapper), Gaetano Salvemini (anti-fascist politician and writer), and Vitangelo Spadavecchia (goalkeeper).

Migration
During the times of the mass migration of Italians, mainly following World War II, many Molfettese residents migrated to Fremantle in Western Australia and to a town in South Australia called Port Pirie. The culture of Molfetta is celebrated in Fremantle and Port Pirie. Officials of both Port Pirie and Molfetta have close links today. In the United States, many Molfettese immigrants settled in the city of Hoboken, New Jersey, where a substantial enclave still exists today.

title: Molfetta


text:
Feline

Feline may refer to:

Zoology

 * Loosely, Felidae, a member of the cat family, which includes the subfamilies Pantherinae and Felinae (conventionally designated a felid).
 * Following the taxonomic convention, Felinae, the subfamily of Felidae that includes domestic cats and smaller wild cats.
 * Cat, the domesticated feline.

Music

 * Feline (band), a late-1990s London-based English rock group.
 * Feline (The Stranglers album), 1983.
 * Feline (1998 album), the self-titled album by Feline, expanded release of Save Your Face.
 * Feline (Ella Eyre album) (2015), the debut studio album by Ella Eyre.
 * Feline (song), a song on Delta Goodrem's 2016 album Wings of the Wild.
 * "Feline", a song by the Cat Empire

Comics

 * Feline (comics), a fictional character from the Malibu comics line.

title: Feline


text:
Pope Paschal II

Pope Paschal II (Paschalis II; 1050 x 1055 – 21 January 1118), born Ranierius, was head of the Catholic Church and ruler of the Papal States from 13 August 1099 to his death in 1118. A monk of the Abbey of Cluny, he was created the cardinal-priest of San Clemente by Pope Gregory VII (1073–85) in 1073. He was consecrated as pope in succession to Pope Urban II (1088–99) on 19 August 1099. His reign of almost twenty years was exceptionally long for a medieval pope.

Early career
Ranierius was born in Bleda, near Forlì, Romagna. He became a monk at Cluny at an early age.

Papacy
After Pope Urban II's death, Paschal reacted to the success of the First Crusade by preaching the penitential Crusade of 1101.

During the long struggle of the papacy with the Holy Roman emperors over investiture, Paschal II zealously carried on the Hildebrandine policy in favor of papal privilege, but with only partial success. Henry V, son of Emperor Henry IV, took advantage of his father's excommunication to rebel, even to the point of seeking out Paschal II for absolution for associating with his father. But, Henry V was even more persistent in maintaining the right of investiture than Emperor Henry IV had been before his death in 1106. The imperial Diet at Mainz invited Paschal II to visit Germany and settle the trouble in January 1106, but the pope in the Council of Guastalla (October 1106) simply renewed the prohibition of investiture.

In the same year he brought to an end the investiture struggle in England, in which Anselm, archbishop of Canterbury, had been engaged with King Henry I, by retaining to himself exclusive right to invest with the ring and crozier, but recognizing the royal nomination to vacate benefices and the oath of fealty for temporal domains. Paschal went to France at the close of 1106 to seek the mediation of King Philip I and his son Louis in the imperial struggle, but he returned to Italy in September 1107, his negotiations remaining without result. When Henry V advanced with an army into Italy in order to be crowned, the pope agreed to a compact in February 1111 which stipulated that before receiving the imperial crown, Henry was to abjure all claims to investitures, whilst the pope undertook to compel the prelates and abbots of the empire to restore all the temporal rights and privileges which they held from the crown. Preparations were made for the coronation on 12 February 1111, but the Romans rose in revolt against Henry, and the German king retired, taking the pope and Curia with him.

After 61 days of harsh imprisonment, during which Prince Robert I of Capua's Norman army was repulsed on its rescue mission, Paschal II yielded and guaranteed investiture to the emperor. Henry V was then crowned in St. Peter's on 13 April 1111, and, after exacting a promise that no revenge would be taken for what had happened, withdrew beyond the Alps. The Hildebrandine party was aroused to action, however; a Lateran council of March 1112 declared null and void the concessions extorted by violence; a council held at Vienne in October 1111 excommunicated the emperor; and Paschal II sanctioned the proceeding.

Pope Paschal II ordered the building of the basilica of Santi Quattro Coronati on the ashes of the one burned during the Norman sack of Rome in 1084.

During Paschal's trip to France in 1106–1107, he consecrated the Cluniac church of Notre Dame at La Charité-sur-Loire, the second largest church in Europe at the time.

Towards the end of his pontificate trouble began anew in England; Paschal II complained in 1115 that councils were held and bishops translated without his authorization, and he threatened Henry I with excommunication. Matilda of Tuscany was said to have bequeathed all her allodial lands to the Church upon her death in 1115, but the donation was neither publicly acknowledged in Rome nor is any documentary record of the donation preserved. Emperor Henry V at once laid claim to Matilda's lands as imperial fiefs and forced the pope to flee from Rome. Paschal II returned after the emperor's withdrawal at the beginning of 1118, but died within a few days, on 21 January 1118.

In 1116, Paschal II, at the behest of Count Ramon Berenguer III of Barcelona, issued a crusade for the capture of Tarragona.

During Paschal's papacy some efforts were made by the Byzantine Emperor Alexios I to bridge the schism between the Orthodox and the Catholic Church, but these failed, as Paschal pressed the demand that the patriarch of Constantinople recognise the pope's primacy over "all the churches of God throughout the world" in late 1112. This was something Orthodox Patriarchs Nicholas Grammaticus and John Agapetus could not do in face of opposition from the majority of clergy, the monastic world, and the laity.

Pope Paschal II issued the bull Pie postulatio voluntatis on 15 February 1113. It brought under Papal protection and confirmed as a religious order the Hospital of Saint John of Jerusalem, later known as the Knights Hospitaller and today known as the Sovereign Military Order of Malta. It also confirmed the order's acquisitions and donations in Europe and Asia and exempted it from all authority save that of the pope.

title: Pope Paschal II